Theorem tz7.49c 8379

Description: Corollary of Proposition 7.49 of [TakeutiZaring] p. 51. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 19-Jan-2013.)

Hypothesis

Ref	Expression
tz7.49c.1	⊢ 𝐹 Fn On

Assertion

Ref	Expression
tz7.49c	⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))) → ∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem tz7.49c

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tz7.49c.1	. . 3 ⊢ 𝐹 Fn On
2		biid 261	. . 3 ⊢ (∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))) ↔ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥))))
3	1, 2	tz7.49 8378	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))) → ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥)))
4		3simpc 1151	. . . 4 ⊢ ((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥)) → ((𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥)))
5		onss 7732	. . . . . . . . 9 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
6		fnssres 6616	. . . . . . . . 9 ⊢ ((𝐹 Fn On ∧ 𝑥 ⊆ On) → (𝐹 ↾ 𝑥) Fn 𝑥)
7	1, 5, 6	sylancr 588	. . . . . . . 8 ⊢ (𝑥 ∈ On → (𝐹 ↾ 𝑥) Fn 𝑥)
8		df-ima 5638	. . . . . . . . . 10 ⊢ (𝐹 “ 𝑥) = ran (𝐹 ↾ 𝑥)
9	8	eqeq1i 2742	. . . . . . . . 9 ⊢ ((𝐹 “ 𝑥) = 𝐴 ↔ ran (𝐹 ↾ 𝑥) = 𝐴)
10	9	biimpi 216	. . . . . . . 8 ⊢ ((𝐹 “ 𝑥) = 𝐴 → ran (𝐹 ↾ 𝑥) = 𝐴)
11	7, 10	anim12i 614	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ (𝐹 “ 𝑥) = 𝐴) → ((𝐹 ↾ 𝑥) Fn 𝑥 ∧ ran (𝐹 ↾ 𝑥) = 𝐴))
12	11	anim1i 616	. . . . . 6 ⊢ (((𝑥 ∈ On ∧ (𝐹 “ 𝑥) = 𝐴) ∧ Fun ◡(𝐹 ↾ 𝑥)) → (((𝐹 ↾ 𝑥) Fn 𝑥 ∧ ran (𝐹 ↾ 𝑥) = 𝐴) ∧ Fun ◡(𝐹 ↾ 𝑥)))
13		dff1o2 6780	. . . . . . 7 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 ↔ ((𝐹 ↾ 𝑥) Fn 𝑥 ∧ Fun ◡(𝐹 ↾ 𝑥) ∧ ran (𝐹 ↾ 𝑥) = 𝐴))
14		3anan32 1097	. . . . . . 7 ⊢ (((𝐹 ↾ 𝑥) Fn 𝑥 ∧ Fun ◡(𝐹 ↾ 𝑥) ∧ ran (𝐹 ↾ 𝑥) = 𝐴) ↔ (((𝐹 ↾ 𝑥) Fn 𝑥 ∧ ran (𝐹 ↾ 𝑥) = 𝐴) ∧ Fun ◡(𝐹 ↾ 𝑥)))
15	13, 14	bitri 275	. . . . . 6 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 ↔ (((𝐹 ↾ 𝑥) Fn 𝑥 ∧ ran (𝐹 ↾ 𝑥) = 𝐴) ∧ Fun ◡(𝐹 ↾ 𝑥)))
16	12, 15	sylibr 234	. . . . 5 ⊢ (((𝑥 ∈ On ∧ (𝐹 “ 𝑥) = 𝐴) ∧ Fun ◡(𝐹 ↾ 𝑥)) → (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴)
17	16	expl 457	. . . 4 ⊢ (𝑥 ∈ On → (((𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥)) → (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴))
18	4, 17	syl5 34	. . 3 ⊢ (𝑥 ∈ On → ((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥)) → (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴))
19	18	reximia 3072	. 2 ⊢ (∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥)) → ∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴)
20	3, 19	syl 17	1 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ On ((𝐴 ∖ (𝐹 “ 𝑥)) ≠ ∅ → (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)))) → ∃𝑥 ∈ On (𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3061   ∖ cdif 3899   ⊆ wss 3902  ∅c0 4286  ^◡ccnv 5624  ran crn 5626   ↾ cres 5627   “ cima 5628  Oncon0 6318  Fun wfun 6487   Fn wfn 6488  –1-1-onto→wf1o 6492  ‘cfv 6493

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 5225  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7682

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-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501

This theorem is referenced by:  dfac8alem  9943  dnnumch1  43322