Theorem tz7.48-3 8467

Description: Proposition 7.48(3) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.)

Hypothesis

Ref	Expression
tz7.48.1	⊢ 𝐹 Fn On

Assertion

Ref	Expression
tz7.48-3	⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ 𝐴 ∈ V)

Distinct variable groups:   𝑥,𝐹   𝑥,𝐴

Proof of Theorem tz7.48-3

Step	Hyp	Ref	Expression
1		tz7.48.1	. . . . 5 ⊢ 𝐹 Fn On
2	1	fndmi 6653	. . . 4 ⊢ dom 𝐹 = On
3		onprc 7781	. . . 4 ⊢ ¬ On ∈ V
4	2, 3	eqneltri 2852	. . 3 ⊢ ¬ dom 𝐹 ∈ V
5	1	tz7.48-2 8465	. . . 4 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → Fun ◡𝐹)
6		funrnex 7961	. . . . . 6 ⊢ (dom ◡𝐹 ∈ V → (Fun ◡𝐹 → ran ◡𝐹 ∈ V))
7	6	com12 32	. . . . 5 ⊢ (Fun ◡𝐹 → (dom ◡𝐹 ∈ V → ran ◡𝐹 ∈ V))
8		df-rn 5678	. . . . . 6 ⊢ ran 𝐹 = dom ◡𝐹
9	8	eleq1i 2824	. . . . 5 ⊢ (ran 𝐹 ∈ V ↔ dom ◡𝐹 ∈ V)
10		dfdm4 5888	. . . . . 6 ⊢ dom 𝐹 = ran ◡𝐹
11	10	eleq1i 2824	. . . . 5 ⊢ (dom 𝐹 ∈ V ↔ ran ◡𝐹 ∈ V)
12	7, 9, 11	3imtr4g 296	. . . 4 ⊢ (Fun ◡𝐹 → (ran 𝐹 ∈ V → dom 𝐹 ∈ V))
13	5, 12	syl 17	. . 3 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → (ran 𝐹 ∈ V → dom 𝐹 ∈ V))
14	4, 13	mtoi 199	. 2 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ ran 𝐹 ∈ V)
15	1	tz7.48-1 8466	. . 3 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ran 𝐹 ⊆ 𝐴)
16		ssexg 5305	. . . 4 ⊢ ((ran 𝐹 ⊆ 𝐴 ∧ 𝐴 ∈ V) → ran 𝐹 ∈ V)
17	16	ex 412	. . 3 ⊢ (ran 𝐹 ⊆ 𝐴 → (𝐴 ∈ V → ran 𝐹 ∈ V))
18	15, 17	syl 17	. 2 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → (𝐴 ∈ V → ran 𝐹 ∈ V))
19	14, 18	mtod 198	1 ⊢ (∀𝑥 ∈ On (𝐹‘𝑥) ∈ (𝐴 ∖ (𝐹 “ 𝑥)) → ¬ 𝐴 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2107  ∀wral 3050  Vcvv 3464   ∖ cdif 3930   ⊆ wss 3933  ^◡ccnv 5666  dom cdm 5667  ran crn 5668   “ cima 5670  Oncon0 6365  Fun wfun 6536   Fn wfn 6537  ‘cfv 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5261  ax-sep 5278  ax-nul 5288  ax-pr 5414  ax-un 7738

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3773  df-csb 3882  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-pss 3953  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-iun 4975  df-br 5126  df-opab 5188  df-mpt 5208  df-tr 5242  df-id 5560  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-ord 6368  df-on 6369  df-iota 6495  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550

This theorem is referenced by:  tz7.49  8468