derangenlem - Mathbox for Mario Carneiro

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Theorem derangenlem 32418

Description: One half of derangen 32419. (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 485	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐴 ≈ 𝐵)
2		bren 8518	. . . . 5 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑠 𝑠:𝐴–1-1-onto→𝐵)
3	1, 2	sylib 220	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → ∃𝑠 𝑠:𝐴–1-1-onto→𝐵)
4		deranglem 32413	. . . . 5 ⊢ (𝐵 ∈ Fin → {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝑓‘𝑦) ≠ 𝑦)} ∈ Fin)
5	4	adantl 484	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → {𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝑓‘𝑦) ≠ 𝑦)} ∈ Fin)
6		f1oco 6637	. . . . . . . . . . . 12 ⊢ ((𝑠:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐴–1-1-onto→𝐴) → (𝑠 ∘ 𝑔):𝐴–1-1-onto→𝐵)
7	6	ad2ant2lr 746	. . . . . . . . . . 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 6627	. . . . . . . . . . . 12 ⊢ (𝑠:𝐴–1-1-onto→𝐵 → ^◡𝑠:𝐵–1-1-onto→𝐴)
9	8	ad2antlr 725	. . . . . . . . . . 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 6637	. . . . . . . . . . 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 586	. . . . . . . . . 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 6118	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∘ 𝑔) ∘ ^◡𝑠) = (𝑠 ∘ (𝑔 ∘ ^◡𝑠))
13	12	fveq1i 6671	. . . . . . . . . . . . . 14 ⊢ (((𝑠 ∘ 𝑔) ∘ ^◡𝑠)‘𝑧) = ((𝑠 ∘ (𝑔 ∘ ^◡𝑠))‘𝑧)
14		simprl 769	. . . . . . . . . . . . . . . . 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 6637	. . . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . . 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 6615	. . . . . . . . . . . . . . . 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 6760	. . . . . . . . . . . . . . 15 ⊢ (((𝑔 ∘ ^◡𝑠):𝐵⟶𝐴 ∧ 𝑧 ∈ 𝐵) → ((𝑠 ∘ (𝑔 ∘ ^◡𝑠))‘𝑧) = (𝑠‘((𝑔 ∘ ^◡𝑠)‘𝑧)))
20	18, 19	sylan 582	. . . . . . . . . . . . . 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	syl5eq 2868	. . . . . . . . . . . . 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 6615	. . . . . . . . . . . . . . . . . 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 6760	. . . . . . . . . . . . . . . . 17 ⊢ ((^◡𝑠:𝐵⟶𝐴 ∧ 𝑧 ∈ 𝐵) → ((𝑔 ∘ ^◡𝑠)‘𝑧) = (𝑔‘(^◡𝑠‘𝑧)))
25	23, 24	sylan 582	. . . . . . . . . . . . . . . 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	ffvelrnda 6851	. . . . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . . 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 6670	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (^◡𝑠‘𝑧) → (𝑔‘𝑦) = (𝑔‘(^◡𝑠‘𝑧)))
29		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (^◡𝑠‘𝑧) → 𝑦 = (^◡𝑠‘𝑧))
30	28, 29	neeq12d 3077	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (^◡𝑠‘𝑧) → ((𝑔‘𝑦) ≠ 𝑦 ↔ (𝑔‘(^◡𝑠‘𝑧)) ≠ (^◡𝑠‘𝑧)))
31	30	rspcv 3618	. . . . . . . . . . . . . . . . 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 3083	. . . . . . . . . . . . . . 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 3071	. . . . . . . . . . . . . 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 774	. . . . . . . . . . . . . . . 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	ffvelrnda 6851	. . . . . . . . . . . . . . . 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 7035	. . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . 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 3037	. . . . . . . . . . . . . 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 3083	. . . . . . . . . . . 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 3182	. . . . . . . . . . 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 6670	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑦 → (((𝑠 ∘ 𝑔) ∘ ^◡𝑠)‘𝑧) = (((𝑠 ∘ 𝑔) ∘ ^◡𝑠)‘𝑦))
44		id 22	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑦 → 𝑧 = 𝑦)
45	43, 44	neeq12d 3077	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → ((((𝑠 ∘ 𝑔) ∘ ^◡𝑠)‘𝑧) ≠ 𝑧 ↔ (((𝑠 ∘ 𝑔) ∘ ^◡𝑠)‘𝑦) ≠ 𝑦))
46	45	cbvralvw 3449	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ 𝐵 (((𝑠 ∘ 𝑔) ∘ ^◡𝑠)‘𝑧) ≠ 𝑧 ↔ ∀𝑦 ∈ 𝐵 (((𝑠 ∘ 𝑔) ∘ ^◡𝑠)‘𝑦) ≠ 𝑦)
47	42, 46	sylib 220	. . . . . . . . . 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 514	. . . . . . . . 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 415	. . . . . . . 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 3497	. . . . . . . . 9 ⊢ 𝑔 ∈ V
51		f1oeq1 6604	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (𝑓:𝐴–1-1-onto→𝐴 ↔ 𝑔:𝐴–1-1-onto→𝐴))
52		fveq1 6669	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑦) = (𝑔‘𝑦))
53	52	neeq1d 3075	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑦) ≠ 𝑦 ↔ (𝑔‘𝑦) ≠ 𝑦))
54	53	ralbidv 3197	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ≠ 𝑦))
55	51, 54	anbi12d 632	. . . . . . . . 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 3667	. . . . . . . 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 3497	. . . . . . . . . . 11 ⊢ 𝑠 ∈ V
58	57, 50	coex 7635	. . . . . . . . . 10 ⊢ (𝑠 ∘ 𝑔) ∈ V
59	57	cnvex 7630	. . . . . . . . . 10 ⊢ ^◡𝑠 ∈ V
60	58, 59	coex 7635	. . . . . . . . 9 ⊢ ((𝑠 ∘ 𝑔) ∘ ^◡𝑠) ∈ V
61		f1oeq1 6604	. . . . . . . . . 10 ⊢ (𝑓 = ((𝑠 ∘ 𝑔) ∘ ^◡𝑠) → (𝑓:𝐵–1-1-onto→𝐵 ↔ ((𝑠 ∘ 𝑔) ∘ ^◡𝑠):𝐵–1-1-onto→𝐵))
62		fveq1 6669	. . . . . . . . . . . 12 ⊢ (𝑓 = ((𝑠 ∘ 𝑔) ∘ ^◡𝑠) → (𝑓‘𝑦) = (((𝑠 ∘ 𝑔) ∘ ^◡𝑠)‘𝑦))
63	62	neeq1d 3075	. . . . . . . . . . 11 ⊢ (𝑓 = ((𝑠 ∘ 𝑔) ∘ ^◡𝑠) → ((𝑓‘𝑦) ≠ 𝑦 ↔ (((𝑠 ∘ 𝑔) ∘ ^◡𝑠)‘𝑦) ≠ 𝑦))
64	63	ralbidv 3197	. . . . . . . . . 10 ⊢ (𝑓 = ((𝑠 ∘ 𝑔) ∘ ^◡𝑠) → (∀𝑦 ∈ 𝐵 (𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ 𝐵 (((𝑠 ∘ 𝑔) ∘ ^◡𝑠)‘𝑦) ≠ 𝑦))
65	61, 64	anbi12d 632	. . . . . . . . 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 3667	. . . . . . . 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 298	. . . . . . 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 3497	. . . . . . . . . 10 ⊢ ℎ ∈ V
69		f1oeq1 6604	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → (𝑓:𝐴–1-1-onto→𝐴 ↔ ℎ:𝐴–1-1-onto→𝐴))
70		fveq1 6669	. . . . . . . . . . . . 13 ⊢ (𝑓 = ℎ → (𝑓‘𝑦) = (ℎ‘𝑦))
71	70	neeq1d 3075	. . . . . . . . . . . 12 ⊢ (𝑓 = ℎ → ((𝑓‘𝑦) ≠ 𝑦 ↔ (ℎ‘𝑦) ≠ 𝑦))
72	71	ralbidv 3197	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → (∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ 𝐴 (ℎ‘𝑦) ≠ 𝑦))
73	69, 72	anbi12d 632	. . . . . . . . . 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 3667	. . . . . . . . 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 628	. . . . . . . 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 725	. . . . . . . . . . . 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 6622	. . . . . . . . . . . 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 715	. . . . . . . . . . . 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 6616	. . . . . . . . . . . 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 767	. . . . . . . . . . . . 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 779	. . . . . . . . . . . . 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 6637	. . . . . . . . . . . . 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 586	. . . . . . . . . . . 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 6616	. . . . . . . . . . . 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 7048	. . . . . . . . . . 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 1367	. . . . . . . . . 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 6614	. . . . . . . . . . . 12 ⊢ (𝑠:𝐴–1-1-onto→𝐵 → 𝑠:𝐴–1-1→𝐵)
91	90	ad2antlr 725	. . . . . . . . . . 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 777	. . . . . . . . . . . 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 6615	. . . . . . . . . . . 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 6615	. . . . . . . . . . . 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 7047	. . . . . . . . . . 11 ⊢ ((𝑠:𝐴–1-1→𝐵 ∧ 𝑔:𝐴⟶𝐴 ∧ ℎ:𝐴⟶𝐴) → ((𝑠 ∘ 𝑔) = (𝑠 ∘ ℎ) ↔ 𝑔 = ℎ))
98	91, 94, 96, 97	syl3anc 1367	. . . . . . . . . 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 281	. . . . . . . . 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 415	. . . . . . . 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	syl5bi 244	. . . . . . 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 8550	. . . . . 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 415	. . . . 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 1934	. . . 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 8735	. . . . . 6 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin)
107	106	ancoms 461	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐴 ∈ Fin)
108		deranglem 32413	. . . . 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 13741	. . . 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 586	. . 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 259	. 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 32414	. . 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 32414	. . 3 ⊢ (𝐵 ∈ Fin → (𝐷‘𝐵) = (♯‘{𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝑓‘𝑦) ≠ 𝑦)}))
117	116	adantl 484	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐷‘𝐵) = (♯‘{𝑓 ∣ (𝑓:𝐵–1-1-onto→𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝑓‘𝑦) ≠ 𝑦)}))
118	112, 115, 117	3brtr4d 5098	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐷‘𝐴) ≤ (𝐷‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 {cab 2799 ≠ wne 3016 ∀wral 3138 class class class wbr 5066 ↦ cmpt 5146 ^◡ccnv 5554 ∘ ccom 5559 Fn wfn 6350 ⟶wf 6351 –1-1→wf1 6352 –onto→wfo 6353 –1-1-onto→wf1o 6354 ‘cfv 6355 ≈ cen 8506 ≼ cdom 8507 Fincfn 8509 ≤ cle 10676 ♯chash 13691

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-xnn0 11969 df-z 11983 df-uz 12245 df-fz 12894 df-hash 13692

This theorem is referenced by: derangen 32419

W3C validator