Theorem tz7.49c 8485

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 8484	. 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 1149	. . . 4 ⊢ ((∀𝑦 ∈ 𝑥 (𝐴 ∖ (𝐹 “ 𝑦)) ≠ ∅ ∧ (𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥)) → ((𝐹 “ 𝑥) = 𝐴 ∧ Fun ◡(𝐹 ↾ 𝑥)))
5		onss 7804	. . . . . . . . 9 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
6		fnssres 6692	. . . . . . . . 9 ⊢ ((𝐹 Fn On ∧ 𝑥 ⊆ On) → (𝐹 ↾ 𝑥) Fn 𝑥)
7	1, 5, 6	sylancr 587	. . . . . . . 8 ⊢ (𝑥 ∈ On → (𝐹 ↾ 𝑥) Fn 𝑥)
8		df-ima 5702	. . . . . . . . . 10 ⊢ (𝐹 “ 𝑥) = ran (𝐹 ↾ 𝑥)
9	8	eqeq1i 2740	. . . . . . . . 9 ⊢ ((𝐹 “ 𝑥) = 𝐴 ↔ ran (𝐹 ↾ 𝑥) = 𝐴)
10	9	biimpi 216	. . . . . . . 8 ⊢ ((𝐹 “ 𝑥) = 𝐴 → ran (𝐹 ↾ 𝑥) = 𝐴)
11	7, 10	anim12i 613	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ (𝐹 “ 𝑥) = 𝐴) → ((𝐹 ↾ 𝑥) Fn 𝑥 ∧ ran (𝐹 ↾ 𝑥) = 𝐴))
12	11	anim1i 615	. . . . . 6 ⊢ (((𝑥 ∈ On ∧ (𝐹 “ 𝑥) = 𝐴) ∧ Fun ◡(𝐹 ↾ 𝑥)) → (((𝐹 ↾ 𝑥) Fn 𝑥 ∧ ran (𝐹 ↾ 𝑥) = 𝐴) ∧ Fun ◡(𝐹 ↾ 𝑥)))
13		dff1o2 6854	. . . . . . 7 ⊢ ((𝐹 ↾ 𝑥):𝑥–1-1-onto→𝐴 ↔ ((𝐹 ↾ 𝑥) Fn 𝑥 ∧ Fun ◡(𝐹 ↾ 𝑥) ∧ ran (𝐹 ↾ 𝑥) = 𝐴))
14		3anan32 1096	. . . . . . 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 3079	. 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 1086   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  ∃wrex 3068   ∖ cdif 3960   ⊆ wss 3963  ∅c0 4339  ^◡ccnv 5688  ran crn 5690   ↾ cres 5691   “ cima 5692  Oncon0 6386  Fun wfun 6557   Fn wfn 6558  –1-1-onto→wf1o 6562  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571

This theorem is referenced by:  dfac8alem  10067  dnnumch1  43033