derangenlem - Mathbox for Mario Carneiro

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Theorem derangenlem 35387

Description: One half of derangen 35388. (Contributed by Mario Carneiro, 22-Jan-2015.)

**Hypothesis**
Ref	Expression
derang.d	⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))

**Assertion**
Ref	Expression
derangenlem	⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐷‘𝐴) ≤ (𝐷‘𝐵))

Distinct variable groups: 𝑥,𝑓,𝑦,𝐴 𝐵,𝑓,𝑥,𝑦 Allowed substitution hints: 𝐷(𝑥,𝑦,𝑓)

Proof of Theorem derangenlem Dummy variables 𝑔 ℎ 𝑠 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 482	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐴 ≈ 𝐵)
2		bren 8905	. . . . 5 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑠 𝑠:𝐴–1-1-onto→𝐵)
3	1, 2	sylib 218	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → ∃𝑠 𝑠:𝐴–1-1-onto→𝐵)
4		deranglem 35382	. . . . 5 ⊢ (𝐵 ∈ Fin → {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝑓‘𝑦) ≠ 𝑦)} ∈ Fin)
5	4	adantl 481	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝑓‘𝑦) ≠ 𝑦)} ∈ Fin)
6		f1oco 6805	. . . . . . . . . . . 12 ⊢ ((𝑠:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐴–1-1-onto→𝐴) → (𝑠 ∘ 𝑔):𝐴–1-1-onto→𝐵)
7	6	ad2ant2lr 749	. . . . . . . . . . 11 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ (𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦)) → (𝑠 ∘ 𝑔):𝐴–1-1-onto→𝐵)
8		f1ocnv 6794	. . . . . . . . . . . 12 ⊢ (𝑠:𝐴–1-1-onto→𝐵 → ^◡𝑠:𝐵–1-1-onto→𝐴)
9	8	ad2antlr 728	. . . . . . . . . . 11 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ (𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦)) → ^◡𝑠:𝐵–1-1-onto→𝐴)
10		f1oco 6805	. . . . . . . . . . 11 ⊢ (((𝑠 ∘ 𝑔):𝐴–1-1-onto→𝐵 ∧ ^◡𝑠:𝐵–1-1-onto→𝐴) → ((𝑠 ∘ 𝑔) ∘ ^◡𝑠):𝐵–1-1-onto→𝐵)
11	7, 9, 10	syl2anc 585	. . . . . . . . . 10 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ (𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦)) → ((𝑠 ∘ 𝑔) ∘ ^◡𝑠):𝐵–1-1-onto→𝐵)
12		coass 6232	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∘ 𝑔) ∘ ^◡𝑠) = (𝑠 ∘ (𝑔 ∘ ^◡𝑠))
13	12	fveq1i 6843	. . . . . . . . . . . . . 14 ⊢ (((𝑠 ∘ 𝑔) ∘ ^◡𝑠)‘𝑧) = ((𝑠 ∘ (𝑔 ∘ ^◡𝑠))‘𝑧)
14		simprl 771	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ (𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦)) → 𝑔:𝐴–1-1-onto→𝐴)
15		f1oco 6805	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔:𝐴–1-1-onto→𝐴 ∧ ^◡𝑠:𝐵–1-1-onto→𝐴) → (𝑔 ∘ ^◡𝑠):𝐵–1-1-onto→𝐴)
16	14, 9, 15	syl2anc 585	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ (𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦)) → (𝑔 ∘ ^◡𝑠):𝐵–1-1-onto→𝐴)
17		f1of 6782	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 ∘ ^◡𝑠):𝐵–1-1-onto→𝐴 → (𝑔 ∘ ^◡𝑠):𝐵⟶𝐴)
18	16, 17	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ (𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦)) → (𝑔 ∘ ^◡𝑠):𝐵⟶𝐴)
19		fvco3 6941	. . . . . . . . . . . . . . 15 ⊢ (((𝑔 ∘ ^◡𝑠):𝐵⟶𝐴 ∧ 𝑧 ∈ 𝐵) → ((𝑠 ∘ (𝑔 ∘ ^◡𝑠))‘𝑧) = (𝑠‘((𝑔 ∘ ^◡𝑠)‘𝑧)))
20	18, 19	sylan 581	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ (𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦)) ∧ 𝑧 ∈ 𝐵) → ((𝑠 ∘ (𝑔 ∘ ^◡𝑠))‘𝑧) = (𝑠‘((𝑔 ∘ ^◡𝑠)‘𝑧)))
21	13, 20	eqtrid 2784	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ (𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦)) ∧ 𝑧 ∈ 𝐵) → (((𝑠 ∘ 𝑔) ∘ ^◡𝑠)‘𝑧) = (𝑠‘((𝑔 ∘ ^◡𝑠)‘𝑧)))
22		f1of 6782	. . . . . . . . . . . . . . . . . 18 ⊢ (^◡𝑠:𝐵–1-1-onto→𝐴 → ^◡𝑠:𝐵⟶𝐴)
23	9, 22	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ (𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦)) → ^◡𝑠:𝐵⟶𝐴)
24		fvco3 6941	. . . . . . . . . . . . . . . . 17 ⊢ ((^◡𝑠:𝐵⟶𝐴 ∧ 𝑧 ∈ 𝐵) → ((𝑔 ∘ ^◡𝑠)‘𝑧) = (𝑔‘(^◡𝑠‘𝑧)))
25	23, 24	sylan 581	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ (𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦)) ∧ 𝑧 ∈ 𝐵) → ((𝑔 ∘ ^◡𝑠)‘𝑧) = (𝑔‘(^◡𝑠‘𝑧)))
26	23	ffvelcdmda 7038	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ (𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦)) ∧ 𝑧 ∈ 𝐵) → (^◡𝑠‘𝑧) ∈ 𝐴)
27		simplrr 778	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ (𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦)) ∧ 𝑧 ∈ 𝐵) → ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦)
28		fveq2 6842	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (^◡𝑠‘𝑧) → (𝑔‘𝑦) = (𝑔‘(^◡𝑠‘𝑧)))
29		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (^◡𝑠‘𝑧) → 𝑦 = (^◡𝑠‘𝑧))
30	28, 29	neeq12d 2994	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (^◡𝑠‘𝑧) → ((𝑔‘𝑦) ≠ 𝑦 ↔ (𝑔‘(^◡𝑠‘𝑧)) ≠ (^◡𝑠‘𝑧)))
31	30	rspcv 3574	. . . . . . . . . . . . . . . . 17 ⊢ ((^◡𝑠‘𝑧) ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦 → (𝑔‘(^◡𝑠‘𝑧)) ≠ (^◡𝑠‘𝑧)))
32	26, 27, 31	sylc 65	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ (𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦)) ∧ 𝑧 ∈ 𝐵) → (𝑔‘(^◡𝑠‘𝑧)) ≠ (^◡𝑠‘𝑧))
33	25, 32	eqnetrd 3000	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ (𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦)) ∧ 𝑧 ∈ 𝐵) → ((𝑔 ∘ ^◡𝑠)‘𝑧) ≠ (^◡𝑠‘𝑧))
34	33	necomd 2988	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ (𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦)) ∧ 𝑧 ∈ 𝐵) → (^◡𝑠‘𝑧) ≠ ((𝑔 ∘ ^◡𝑠)‘𝑧))
35		simpllr 776	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ (𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦)) ∧ 𝑧 ∈ 𝐵) → 𝑠:𝐴–1-1-onto→𝐵)
36	18	ffvelcdmda 7038	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ (𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦)) ∧ 𝑧 ∈ 𝐵) → ((𝑔 ∘ ^◡𝑠)‘𝑧) ∈ 𝐴)
37		f1ocnvfv 7234	. . . . . . . . . . . . . . . 16 ⊢ ((𝑠:𝐴–1-1-onto→𝐵 ∧ ((𝑔 ∘ ^◡𝑠)‘𝑧) ∈ 𝐴) → ((𝑠‘((𝑔 ∘ ^◡𝑠)‘𝑧)) = 𝑧 → (^◡𝑠‘𝑧) = ((𝑔 ∘ ^◡𝑠)‘𝑧)))
38	35, 36, 37	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ (𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦)) ∧ 𝑧 ∈ 𝐵) → ((𝑠‘((𝑔 ∘ ^◡𝑠)‘𝑧)) = 𝑧 → (^◡𝑠‘𝑧) = ((𝑔 ∘ ^◡𝑠)‘𝑧)))
39	38	necon3d 2954	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ (𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦)) ∧ 𝑧 ∈ 𝐵) → ((^◡𝑠‘𝑧) ≠ ((𝑔 ∘ ^◡𝑠)‘𝑧) → (𝑠‘((𝑔 ∘ ^◡𝑠)‘𝑧)) ≠ 𝑧))
40	34, 39	mpd 15	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ (𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦)) ∧ 𝑧 ∈ 𝐵) → (𝑠‘((𝑔 ∘ ^◡𝑠)‘𝑧)) ≠ 𝑧)
41	21, 40	eqnetrd 3000	. . . . . . . . . . . 12 ⊢ (((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ (𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦)) ∧ 𝑧 ∈ 𝐵) → (((𝑠 ∘ 𝑔) ∘ ^◡𝑠)‘𝑧) ≠ 𝑧)
42	41	ralrimiva 3130	. . . . . . . . . . 11 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ (𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦)) → ∀𝑧 ∈ 𝐵 (((𝑠 ∘ 𝑔) ∘ ^◡𝑠)‘𝑧) ≠ 𝑧)
43		fveq2 6842	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑦 → (((𝑠 ∘ 𝑔) ∘ ^◡𝑠)‘𝑧) = (((𝑠 ∘ 𝑔) ∘ ^◡𝑠)‘𝑦))
44		id 22	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑦 → 𝑧 = 𝑦)
45	43, 44	neeq12d 2994	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → ((((𝑠 ∘ 𝑔) ∘ ^◡𝑠)‘𝑧) ≠ 𝑧 ↔ (((𝑠 ∘ 𝑔) ∘ ^◡𝑠)‘𝑦) ≠ 𝑦))
46	45	cbvralvw 3216	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ 𝐵 (((𝑠 ∘ 𝑔) ∘ ^◡𝑠)‘𝑧) ≠ 𝑧 ↔ ∀𝑦 ∈ 𝐵 (((𝑠 ∘ 𝑔) ∘ ^◡𝑠)‘𝑦) ≠ 𝑦)
47	42, 46	sylib 218	. . . . . . . . . 10 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ (𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦)) → ∀𝑦 ∈ 𝐵 (((𝑠 ∘ 𝑔) ∘ ^◡𝑠)‘𝑦) ≠ 𝑦)
48	11, 47	jca 511	. . . . . . . . 9 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ (𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦)) → (((𝑠 ∘ 𝑔) ∘ ^◡𝑠):𝐵–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐵 (((𝑠 ∘ 𝑔) ∘ ^◡𝑠)‘𝑦) ≠ 𝑦))
49	48	ex 412	. . . . . . . 8 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) → ((𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦) → (((𝑠 ∘ 𝑔) ∘ ^◡𝑠):𝐵–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐵 (((𝑠 ∘ 𝑔) ∘ ^◡𝑠)‘𝑦) ≠ 𝑦)))
50		vex 3446	. . . . . . . . 9 ⊢ 𝑔 ∈ V
51		f1oeq1 6770	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (𝑓:𝐴–1-1-onto→𝐴 ↔ 𝑔:𝐴–1-1-onto→𝐴))
52		fveq1 6841	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑦) = (𝑔‘𝑦))
53	52	neeq1d 2992	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑦) ≠ 𝑦 ↔ (𝑔‘𝑦) ≠ 𝑦))
54	53	ralbidv 3161	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦))
55	51, 54	anbi12d 633	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → ((𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦) ↔ (𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦)))
56	50, 55	elab 3636	. . . . . . . 8 ⊢ (𝑔 ∈ {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)} ↔ (𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦))
57		vex 3446	. . . . . . . . . . 11 ⊢ 𝑠 ∈ V
58	57, 50	coex 7882	. . . . . . . . . 10 ⊢ (𝑠 ∘ 𝑔) ∈ V
59	57	cnvex 7877	. . . . . . . . . 10 ⊢ ^◡𝑠 ∈ V
60	58, 59	coex 7882	. . . . . . . . 9 ⊢ ((𝑠 ∘ 𝑔) ∘ ^◡𝑠) ∈ V
61		f1oeq1 6770	. . . . . . . . . 10 ⊢ (𝑓 = ((𝑠 ∘ 𝑔) ∘ ^◡𝑠) → (𝑓:𝐵–1-1-onto→𝐵 ↔ ((𝑠 ∘ 𝑔) ∘ ^◡𝑠):𝐵–1-1-onto→𝐵))
62		fveq1 6841	. . . . . . . . . . . 12 ⊢ (𝑓 = ((𝑠 ∘ 𝑔) ∘ ^◡𝑠) → (𝑓‘𝑦) = (((𝑠 ∘ 𝑔) ∘ ^◡𝑠)‘𝑦))
63	62	neeq1d 2992	. . . . . . . . . . 11 ⊢ (𝑓 = ((𝑠 ∘ 𝑔) ∘ ^◡𝑠) → ((𝑓‘𝑦) ≠ 𝑦 ↔ (((𝑠 ∘ 𝑔) ∘ ^◡𝑠)‘𝑦) ≠ 𝑦))
64	63	ralbidv 3161	. . . . . . . . . 10 ⊢ (𝑓 = ((𝑠 ∘ 𝑔) ∘ ^◡𝑠) → (∀𝑦 ∈ 𝐵 (𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ 𝐵 (((𝑠 ∘ 𝑔) ∘ ^◡𝑠)‘𝑦) ≠ 𝑦))
65	61, 64	anbi12d 633	. . . . . . . . 9 ⊢ (𝑓 = ((𝑠 ∘ 𝑔) ∘ ^◡𝑠) → ((𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝑓‘𝑦) ≠ 𝑦) ↔ (((𝑠 ∘ 𝑔) ∘ ^◡𝑠):𝐵–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐵 (((𝑠 ∘ 𝑔) ∘ ^◡𝑠)‘𝑦) ≠ 𝑦)))
66	60, 65	elab 3636	. . . . . . . 8 ⊢ (((𝑠 ∘ 𝑔) ∘ ^◡𝑠) ∈ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝑓‘𝑦) ≠ 𝑦)} ↔ (((𝑠 ∘ 𝑔) ∘ ^◡𝑠):𝐵–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐵 (((𝑠 ∘ 𝑔) ∘ ^◡𝑠)‘𝑦) ≠ 𝑦))
67	49, 56, 66	3imtr4g 296	. . . . . . 7 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) → (𝑔 ∈ {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)} → ((𝑠 ∘ 𝑔) ∘ ^◡𝑠) ∈ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝑓‘𝑦) ≠ 𝑦)}))
68		vex 3446	. . . . . . . . . 10 ⊢ ℎ ∈ V
69		f1oeq1 6770	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → (𝑓:𝐴–1-1-onto→𝐴 ↔ ℎ:𝐴–1-1-onto→𝐴))
70		fveq1 6841	. . . . . . . . . . . . 13 ⊢ (𝑓 = ℎ → (𝑓‘𝑦) = (ℎ‘𝑦))
71	70	neeq1d 2992	. . . . . . . . . . . 12 ⊢ (𝑓 = ℎ → ((𝑓‘𝑦) ≠ 𝑦 ↔ (ℎ‘𝑦) ≠ 𝑦))
72	71	ralbidv 3161	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → (∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ≠ 𝑦))
73	69, 72	anbi12d 633	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → ((𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦) ↔ (ℎ:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ≠ 𝑦)))
74	68, 73	elab 3636	. . . . . . . . 9 ⊢ (ℎ ∈ {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)} ↔ (ℎ:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ≠ 𝑦))
75	56, 74	anbi12i 629	. . . . . . . 8 ⊢ ((𝑔 ∈ {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)} ∧ ℎ ∈ {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)}) ↔ ((𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦) ∧ (ℎ:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ≠ 𝑦)))
76	8	ad2antlr 728	. . . . . . . . . . . 12 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ ((𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦) ∧ (ℎ:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ≠ 𝑦))) → ^◡𝑠:𝐵–1-1-onto→𝐴)
77		f1ofo 6789	. . . . . . . . . . . 12 ⊢ (^◡𝑠:𝐵–1-1-onto→𝐴 → ^◡𝑠:𝐵–onto→𝐴)
78	76, 77	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ ((𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦) ∧ (ℎ:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ≠ 𝑦))) → ^◡𝑠:𝐵–onto→𝐴)
79	7	adantrr 718	. . . . . . . . . . . 12 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ ((𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦) ∧ (ℎ:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ≠ 𝑦))) → (𝑠 ∘ 𝑔):𝐴–1-1-onto→𝐵)
80		f1ofn 6783	. . . . . . . . . . . 12 ⊢ ((𝑠 ∘ 𝑔):𝐴–1-1-onto→𝐵 → (𝑠 ∘ 𝑔) Fn 𝐴)
81	79, 80	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ ((𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦) ∧ (ℎ:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ≠ 𝑦))) → (𝑠 ∘ 𝑔) Fn 𝐴)
82		simplr 769	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ ((𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦) ∧ (ℎ:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ≠ 𝑦))) → 𝑠:𝐴–1-1-onto→𝐵)
83		simprrl 781	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ ((𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦) ∧ (ℎ:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ≠ 𝑦))) → ℎ:𝐴–1-1-onto→𝐴)
84		f1oco 6805	. . . . . . . . . . . . 13 ⊢ ((𝑠:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐴–1-1-onto→𝐴) → (𝑠 ∘ ℎ):𝐴–1-1-onto→𝐵)
85	82, 83, 84	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ ((𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦) ∧ (ℎ:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ≠ 𝑦))) → (𝑠 ∘ ℎ):𝐴–1-1-onto→𝐵)
86		f1ofn 6783	. . . . . . . . . . . 12 ⊢ ((𝑠 ∘ ℎ):𝐴–1-1-onto→𝐵 → (𝑠 ∘ ℎ) Fn 𝐴)
87	85, 86	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ ((𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦) ∧ (ℎ:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ≠ 𝑦))) → (𝑠 ∘ ℎ) Fn 𝐴)
88		cocan2 7248	. . . . . . . . . . 11 ⊢ ((^◡𝑠:𝐵–onto→𝐴 ∧ (𝑠 ∘ 𝑔) Fn 𝐴 ∧ (𝑠 ∘ ℎ) Fn 𝐴) → (((𝑠 ∘ 𝑔) ∘ ^◡𝑠) = ((𝑠 ∘ ℎ) ∘ ^◡𝑠) ↔ (𝑠 ∘ 𝑔) = (𝑠 ∘ ℎ)))
89	78, 81, 87, 88	syl3anc 1374	. . . . . . . . . 10 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ ((𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦) ∧ (ℎ:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ≠ 𝑦))) → (((𝑠 ∘ 𝑔) ∘ ^◡𝑠) = ((𝑠 ∘ ℎ) ∘ ^◡𝑠) ↔ (𝑠 ∘ 𝑔) = (𝑠 ∘ ℎ)))
90		f1of1 6781	. . . . . . . . . . . 12 ⊢ (𝑠:𝐴–1-1-onto→𝐵 → 𝑠:𝐴–1-1→𝐵)
91	90	ad2antlr 728	. . . . . . . . . . 11 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ ((𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦) ∧ (ℎ:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ≠ 𝑦))) → 𝑠:𝐴–1-1→𝐵)
92		simprll 779	. . . . . . . . . . . 12 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ ((𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦) ∧ (ℎ:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ≠ 𝑦))) → 𝑔:𝐴–1-1-onto→𝐴)
93		f1of 6782	. . . . . . . . . . . 12 ⊢ (𝑔:𝐴–1-1-onto→𝐴 → 𝑔:𝐴⟶𝐴)
94	92, 93	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ ((𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦) ∧ (ℎ:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ≠ 𝑦))) → 𝑔:𝐴⟶𝐴)
95		f1of 6782	. . . . . . . . . . . 12 ⊢ (ℎ:𝐴–1-1-onto→𝐴 → ℎ:𝐴⟶𝐴)
96	83, 95	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ ((𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦) ∧ (ℎ:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ≠ 𝑦))) → ℎ:𝐴⟶𝐴)
97		cocan1 7247	. . . . . . . . . . 11 ⊢ ((𝑠:𝐴–1-1→𝐵 ∧ 𝑔:𝐴⟶𝐴 ∧ ℎ:𝐴⟶𝐴) → ((𝑠 ∘ 𝑔) = (𝑠 ∘ ℎ) ↔ 𝑔 = ℎ))
98	91, 94, 96, 97	syl3anc 1374	. . . . . . . . . 10 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ ((𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦) ∧ (ℎ:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ≠ 𝑦))) → ((𝑠 ∘ 𝑔) = (𝑠 ∘ ℎ) ↔ 𝑔 = ℎ))
99	89, 98	bitrd 279	. . . . . . . . 9 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) ∧ ((𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦) ∧ (ℎ:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ≠ 𝑦))) → (((𝑠 ∘ 𝑔) ∘ ^◡𝑠) = ((𝑠 ∘ ℎ) ∘ ^◡𝑠) ↔ 𝑔 = ℎ))
100	99	ex 412	. . . . . . . 8 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) → (((𝑔:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦) ∧ (ℎ:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ≠ 𝑦)) → (((𝑠 ∘ 𝑔) ∘ ^◡𝑠) = ((𝑠 ∘ ℎ) ∘ ^◡𝑠) ↔ 𝑔 = ℎ)))
101	75, 100	biimtrid 242	. . . . . . 7 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) → ((𝑔 ∈ {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)} ∧ ℎ ∈ {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)}) → (((𝑠 ∘ 𝑔) ∘ ^◡𝑠) = ((𝑠 ∘ ℎ) ∘ ^◡𝑠) ↔ 𝑔 = ℎ)))
102	67, 101	dom2d 8942	. . . . . 6 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) ∧ 𝑠:𝐴–1-1-onto→𝐵) → ({𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝑓‘𝑦) ≠ 𝑦)} ∈ Fin → {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)} ≼ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝑓‘𝑦) ≠ 𝑦)}))
103	102	ex 412	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝑠:𝐴–1-1-onto→𝐵 → ({𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝑓‘𝑦) ≠ 𝑦)} ∈ Fin → {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)} ≼ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝑓‘𝑦) ≠ 𝑦)})))
104	103	exlimdv 1935	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (∃𝑠 𝑠:𝐴–1-1-onto→𝐵 → ({𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝑓‘𝑦) ≠ 𝑦)} ∈ Fin → {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)} ≼ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝑓‘𝑦) ≠ 𝑦)})))
105	3, 5, 104	mp2d 49	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)} ≼ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝑓‘𝑦) ≠ 𝑦)})
106		enfii 9122	. . . . . 6 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin)
107	106	ancoms 458	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐴 ∈ Fin)
108		deranglem 35382	. . . . 5 ⊢ (𝐴 ∈ Fin → {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)} ∈ Fin)
109	107, 108	syl 17	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)} ∈ Fin)
110		hashdom 14314	. . . 4 ⊢ (({𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)} ∈ Fin ∧ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝑓‘𝑦) ≠ 𝑦)} ∈ Fin) → ((♯‘{𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)}) ≤ (♯‘{𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝑓‘𝑦) ≠ 𝑦)}) ↔ {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)} ≼ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝑓‘𝑦) ≠ 𝑦)}))
111	109, 5, 110	syl2anc 585	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → ((♯‘{𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)}) ≤ (♯‘{𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝑓‘𝑦) ≠ 𝑦)}) ↔ {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)} ≼ {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝑓‘𝑦) ≠ 𝑦)}))
112	105, 111	mpbird 257	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (♯‘{𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)}) ≤ (♯‘{𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝑓‘𝑦) ≠ 𝑦)}))
113		derang.d	. . . 4 ⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))
114	113	derangval 35383	. . 3 ⊢ (𝐴 ∈ Fin → (𝐷‘𝐴) = (♯‘{𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)}))
115	107, 114	syl 17	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐷‘𝐴) = (♯‘{𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)}))
116	113	derangval 35383	. . 3 ⊢ (𝐵 ∈ Fin → (𝐷‘𝐵) = (♯‘{𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝑓‘𝑦) ≠ 𝑦)}))
117	116	adantl 481	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐷‘𝐵) = (♯‘{𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝑓‘𝑦) ≠ 𝑦)}))
118	112, 115, 117	3brtr4d 5132	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐷‘𝐴) ≤ (𝐷‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∃wex 1781 ∈ wcel 2114 {cab 2715 ≠ wne 2933 ∀wral 3052 class class class wbr 5100 ↦ cmpt 5181 ^◡ccnv 5631 ∘ ccom 5636 Fn wfn 6495 ⟶wf 6496 –1-1→wf1 6497 –onto→wfo 6498 –1-1-onto→wf1o 6499 ‘cfv 6500 ≈ cen 8892 ≼ cdom 8893 Fincfn 8895 ≤ cle 11179 ♯chash 14265

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-oadd 8411 df-er 8645 df-map 8777 df-pm 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-n0 12414 df-xnn0 12487 df-z 12501 df-uz 12764 df-fz 13436 df-hash 14266

This theorem is referenced by: derangen 35388

W3C validator