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Theorem hashdom 14094

Description: Dominance relation for the size function. (Contributed by Mario Carneiro, 22-Sep-2013.) (Revised by Mario Carneiro, 22-Apr-2015.)

**Assertion**
Ref	Expression
hashdom	⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ 𝐴 ≼ 𝐵))

Proof of Theorem hashdom Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzfi 13692	. . . . . . . 8 ⊢ (1...((♯‘𝐵) − (♯‘𝐴))) ∈ Fin
2		ficardom 9719	. . . . . . . 8 ⊢ ((1...((♯‘𝐵) − (♯‘𝐴))) ∈ Fin → (card‘(1...((♯‘𝐵) − (♯‘𝐴)))) ∈ ω)
3	1, 2	ax-mp 5	. . . . . . 7 ⊢ (card‘(1...((♯‘𝐵) − (♯‘𝐴)))) ∈ ω
4		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
5	4	hashgval 14047	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Fin → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) = (♯‘𝐴))
6	5	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) = (♯‘𝐴))
7	4	hashgval 14047	. . . . . . . . . . . . . 14 ⊢ ((1...((♯‘𝐵) − (♯‘𝐴))) ∈ Fin → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...((♯‘𝐵) − (♯‘𝐴))))) = (♯‘(1...((♯‘𝐵) − (♯‘𝐴)))))
8	1, 7	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...((♯‘𝐵) − (♯‘𝐴))))) = (♯‘(1...((♯‘𝐵) − (♯‘𝐴))))
9		hashcl 14071	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
10	9	ad2antrr 723	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (♯‘𝐴) ∈ ℕ₀)
11		hashcl 14071	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ₀)
12	11	ad2antlr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (♯‘𝐵) ∈ ℕ₀)
13		simpr 485	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (♯‘𝐴) ≤ (♯‘𝐵))
14		nn0sub2 12381	. . . . . . . . . . . . . . 15 ⊢ (((♯‘𝐴) ∈ ℕ₀ ∧ (♯‘𝐵) ∈ ℕ₀ ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((♯‘𝐵) − (♯‘𝐴)) ∈ ℕ₀)
15	10, 12, 13, 14	syl3anc 1370	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((♯‘𝐵) − (♯‘𝐴)) ∈ ℕ₀)
16		hashfz1 14060	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐵) − (♯‘𝐴)) ∈ ℕ₀ → (♯‘(1...((♯‘𝐵) − (♯‘𝐴)))) = ((♯‘𝐵) − (♯‘𝐴)))
17	15, 16	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (♯‘(1...((♯‘𝐵) − (♯‘𝐴)))) = ((♯‘𝐵) − (♯‘𝐴)))
18	8, 17	eqtrid 2790	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...((♯‘𝐵) − (♯‘𝐴))))) = ((♯‘𝐵) − (♯‘𝐴)))
19	6, 18	oveq12d 7293	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) + ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...((♯‘𝐵) − (♯‘𝐴)))))) = ((♯‘𝐴) + ((♯‘𝐵) − (♯‘𝐴))))
20	9	nn0cnd 12295	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℂ)
21	11	nn0cnd 12295	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℂ)
22		pncan3 11229	. . . . . . . . . . . . 13 ⊢ (((♯‘𝐴) ∈ ℂ ∧ (♯‘𝐵) ∈ ℂ) → ((♯‘𝐴) + ((♯‘𝐵) − (♯‘𝐴))) = (♯‘𝐵))
23	20, 21, 22	syl2an 596	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) + ((♯‘𝐵) − (♯‘𝐴))) = (♯‘𝐵))
24	23	adantr 481	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((♯‘𝐴) + ((♯‘𝐵) − (♯‘𝐴))) = (♯‘𝐵))
25	19, 24	eqtrd 2778	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) + ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...((♯‘𝐵) − (♯‘𝐴)))))) = (♯‘𝐵))
26		ficardom 9719	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
27	26	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (card‘𝐴) ∈ ω)
28	4	hashgadd 14092	. . . . . . . . . . 11 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘(1...((♯‘𝐵) − (♯‘𝐴)))) ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴)))))) = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) + ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...((♯‘𝐵) − (♯‘𝐴)))))))
29	27, 3, 28	sylancl 586	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴)))))) = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) + ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...((♯‘𝐵) − (♯‘𝐴)))))))
30	4	hashgval 14047	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Fin → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) = (♯‘𝐵))
31	30	ad2antlr 724	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) = (♯‘𝐵))
32	25, 29, 31	3eqtr4d 2788	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴)))))) = ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)))
33	32	fveq2d 6778	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (^◡(rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))))) = (^◡(rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵))))
34	4	hashgf1o 13691	. . . . . . . . 9 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω):ω–1-1-onto→ℕ₀
35		nnacl 8442	. . . . . . . . . 10 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘(1...((♯‘𝐵) − (♯‘𝐴)))) ∈ ω) → ((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))) ∈ ω)
36	27, 3, 35	sylancl 586	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))) ∈ ω)
37		f1ocnvfv1 7148	. . . . . . . . 9 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω):ω–1-1-onto→ℕ₀ ∧ ((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))) ∈ ω) → (^◡(rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))))) = ((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))))
38	34, 36, 37	sylancr 587	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (^◡(rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))))) = ((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))))
39		ficardom 9719	. . . . . . . . . 10 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
40	39	ad2antlr 724	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (card‘𝐵) ∈ ω)
41		f1ocnvfv1 7148	. . . . . . . . 9 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω):ω–1-1-onto→ℕ₀ ∧ (card‘𝐵) ∈ ω) → (^◡(rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵))) = (card‘𝐵))
42	34, 40, 41	sylancr 587	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (^◡(rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵))) = (card‘𝐵))
43	33, 38, 42	3eqtr3d 2786	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))) = (card‘𝐵))
44		oveq2 7283	. . . . . . . . 9 ⊢ (𝑦 = (card‘(1...((♯‘𝐵) − (♯‘𝐴)))) → ((card‘𝐴) +_o 𝑦) = ((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))))
45	44	eqeq1d 2740	. . . . . . . 8 ⊢ (𝑦 = (card‘(1...((♯‘𝐵) − (♯‘𝐴)))) → (((card‘𝐴) +_o 𝑦) = (card‘𝐵) ↔ ((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))) = (card‘𝐵)))
46	45	rspcev 3561	. . . . . . 7 ⊢ (((card‘(1...((♯‘𝐵) − (♯‘𝐴)))) ∈ ω ∧ ((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))) = (card‘𝐵)) → ∃𝑦 ∈ ω ((card‘𝐴) +_o 𝑦) = (card‘𝐵))
47	3, 43, 46	sylancr 587	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ∃𝑦 ∈ ω ((card‘𝐴) +_o 𝑦) = (card‘𝐵))
48	47	ex 413	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) ≤ (♯‘𝐵) → ∃𝑦 ∈ ω ((card‘𝐴) +_o 𝑦) = (card‘𝐵)))
49		cardnn 9721	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (card‘𝑦) = 𝑦)
50	49	adantl 482	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (card‘𝑦) = 𝑦)
51	50	oveq2d 7291	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → ((card‘𝐴) +_o (card‘𝑦)) = ((card‘𝐴) +_o 𝑦))
52	51	eqeq1d 2740	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (((card‘𝐴) +_o (card‘𝑦)) = (card‘𝐵) ↔ ((card‘𝐴) +_o 𝑦) = (card‘𝐵)))
53		fveq2 6774	. . . . . . . 8 ⊢ (((card‘𝐴) +_o (card‘𝑦)) = (card‘𝐵) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘𝑦))) = ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)))
54		nnfi 8950	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → 𝑦 ∈ Fin)
55		ficardom 9719	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ Fin → (card‘𝑦) ∈ ω)
56	4	hashgadd 14092	. . . . . . . . . . . . . 14 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘𝑦) ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘𝑦))) = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) + ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝑦))))
57	26, 55, 56	syl2an 596	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘𝑦))) = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) + ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝑦))))
58	4	hashgval 14047	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ Fin → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝑦)) = (♯‘𝑦))
59	5, 58	oveqan12d 7294	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) + ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝑦))) = ((♯‘𝐴) + (♯‘𝑦)))
60	57, 59	eqtrd 2778	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘𝑦))) = ((♯‘𝐴) + (♯‘𝑦)))
61	60	adantlr 712	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘𝑦))) = ((♯‘𝐴) + (♯‘𝑦)))
62	30	ad2antlr 724	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) = (♯‘𝐵))
63	61, 62	eqeq12d 2754	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘𝑦))) = ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) ↔ ((♯‘𝐴) + (♯‘𝑦)) = (♯‘𝐵)))
64		hashcl 14071	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ Fin → (♯‘𝑦) ∈ ℕ₀)
65	64	nn0ge0d 12296	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ Fin → 0 ≤ (♯‘𝑦))
66	65	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → 0 ≤ (♯‘𝑦))
67	9	nn0red 12294	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℝ)
68	64	nn0red 12294	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ Fin → (♯‘𝑦) ∈ ℝ)
69		addge01 11485	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐴) ∈ ℝ ∧ (♯‘𝑦) ∈ ℝ) → (0 ≤ (♯‘𝑦) ↔ (♯‘𝐴) ≤ ((♯‘𝐴) + (♯‘𝑦))))
70	67, 68, 69	syl2an 596	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → (0 ≤ (♯‘𝑦) ↔ (♯‘𝐴) ≤ ((♯‘𝐴) + (♯‘𝑦))))
71	66, 70	mpbid 231	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → (♯‘𝐴) ≤ ((♯‘𝐴) + (♯‘𝑦)))
72	71	adantlr 712	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → (♯‘𝐴) ≤ ((♯‘𝐴) + (♯‘𝑦)))
73		breq2 5078	. . . . . . . . . . 11 ⊢ (((♯‘𝐴) + (♯‘𝑦)) = (♯‘𝐵) → ((♯‘𝐴) ≤ ((♯‘𝐴) + (♯‘𝑦)) ↔ (♯‘𝐴) ≤ (♯‘𝐵)))
74	72, 73	syl5ibcom 244	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → (((♯‘𝐴) + (♯‘𝑦)) = (♯‘𝐵) → (♯‘𝐴) ≤ (♯‘𝐵)))
75	63, 74	sylbid 239	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘𝑦))) = ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) → (♯‘𝐴) ≤ (♯‘𝐵)))
76	54, 75	sylan2 593	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘𝑦))) = ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) → (♯‘𝐴) ≤ (♯‘𝐵)))
77	53, 76	syl5 34	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (((card‘𝐴) +_o (card‘𝑦)) = (card‘𝐵) → (♯‘𝐴) ≤ (♯‘𝐵)))
78	52, 77	sylbird 259	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (((card‘𝐴) +_o 𝑦) = (card‘𝐵) → (♯‘𝐴) ≤ (♯‘𝐵)))
79	78	rexlimdva 3213	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (∃𝑦 ∈ ω ((card‘𝐴) +_o 𝑦) = (card‘𝐵) → (♯‘𝐴) ≤ (♯‘𝐵)))
80	48, 79	impbid 211	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ ∃𝑦 ∈ ω ((card‘𝐴) +_o 𝑦) = (card‘𝐵)))
81		nnawordex 8468	. . . . 5 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ∃𝑦 ∈ ω ((card‘𝐴) +_o 𝑦) = (card‘𝐵)))
82	26, 39, 81	syl2an 596	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ∃𝑦 ∈ ω ((card‘𝐴) +_o 𝑦) = (card‘𝐵)))
83		finnum 9706	. . . . 5 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card)
84		finnum 9706	. . . . 5 ⊢ (𝐵 ∈ Fin → 𝐵 ∈ dom card)
85		carddom2 9735	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))
86	83, 84, 85	syl2an 596	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))
87	80, 82, 86	3bitr2d 307	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ 𝐴 ≼ 𝐵))
88	87	adantlr 712	. 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ 𝐴 ≼ 𝐵))
89		hashxrcl 14072	. . . . . 6 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℝ^*)
90	89	ad2antrr 723	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐴) ∈ ℝ^*)
91		pnfge 12866	. . . . 5 ⊢ ((♯‘𝐴) ∈ ℝ^* → (♯‘𝐴) ≤ +∞)
92	90, 91	syl 17	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐴) ≤ +∞)
93		hashinf 14049	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐵) = +∞)
94	93	adantll 711	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐵) = +∞)
95	92, 94	breqtrrd 5102	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐴) ≤ (♯‘𝐵))
96		isinffi 9750	. . . . . 6 ⊢ ((¬ 𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → ∃𝑓 𝑓:𝐴–1-1→𝐵)
97	96	ancoms 459	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐴–1-1→𝐵)
98	97	adantlr 712	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐴–1-1→𝐵)
99		brdomg 8746	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
100	99	ad2antlr 724	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
101	98, 100	mpbird 256	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → 𝐴 ≼ 𝐵)
102	95, 101	2thd 264	. 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ 𝐴 ≼ 𝐵))
103	88, 102	pm2.61dan 810	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ 𝐴 ≼ 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∃wex 1782 ∈ wcel 2106 ∃wrex 3065 Vcvv 3432 ⊆ wss 3887 class class class wbr 5074 ↦ cmpt 5157 ^◡ccnv 5588 dom cdm 5589 ↾ cres 5591 –1-1→wf1 6430 –1-1-onto→wf1o 6432 ‘cfv 6433 (class class class)co 7275 ωcom 7712 reccrdg 8240 +_o coa 8294 ≼ cdom 8731 Fincfn 8733 cardccrd 9693 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 +∞cpnf 11006 ℝ^*cxr 11008 ≤ cle 11010 − cmin 11205 ℕ₀cn0 12233 ...cfz 13239 ♯chash 14044

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-oadd 8301 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-xnn0 12306 df-z 12320 df-uz 12583 df-fz 13240 df-hash 14045

This theorem is referenced by: hashdomi 14095 hashsdom 14096 hashun2 14098 hashss 14124 hashsslei 14141 hashfun 14152 hashf1 14171 hashge3el3dif 14200 isercoll 15379 phicl2 16469 phibnd 16472 prmreclem2 16618 prmreclem3 16619 4sqlem11 16656 vdwlem11 16692 ramub2 16715 0ram 16721 ram0 16723 sylow1lem4 19206 pgpssslw 19219 fislw 19230 znfld 20768 znidomb 20769 fta1blem 25333 birthdaylem3 26103 basellem4 26233 ppiwordi 26311 musum 26340 ppiub 26352 chpub 26368 lgsqrlem4 26497 upgrex 27462 sizusglecusg 27830 derangenlem 33133 subfaclefac 33138 erdsze2lem1 33165 snmlff 33291 idomsubgmo 41023 aacllem 46505

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