Theorem permaxun 45720

Description: The Axiom of Union ax-un 7732 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148. (Contributed by Eric Schmidt, 6-Nov-2025.)

Hypotheses

Ref	Expression
permmodel.1	⊢ 𝐹:V–1-1-onto→V
permmodel.2	⊢ 𝑅 = (◡𝐹 ∘ E )

Assertion

Ref	Expression
permaxun	⊢ ∃𝑦∀𝑧(∃𝑤(𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧𝑅𝑦)

Distinct variable groups:   𝑥,𝑤,𝑦,𝑧   𝑤,𝐹,𝑦,𝑧   𝑤,𝑅

Allowed substitution hints:   𝑅(𝑥,𝑦,𝑧)   𝐹(𝑥)

Proof of Theorem permaxun

Step	Hyp	Ref	Expression
1		fvex 6894	. 2 ⊢ (◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))) ∈ V
2		breq2 5113	. . . 4 ⊢ (𝑦 = (◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))) → (𝑧𝑅𝑦 ↔ 𝑧𝑅(◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥)))))
3	2	imbi2d 343	. . 3 ⊢ (𝑦 = (◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))) → ((∃𝑤(𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧𝑅𝑦) ↔ (∃𝑤(𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧𝑅(◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))))))
4	3	albidv 1950	. 2 ⊢ (𝑦 = (◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))) → (∀𝑧(∃𝑤(𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧𝑅𝑦) ↔ ∀𝑧(∃𝑤(𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧𝑅(◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))))))
5		permmodel.1	. . . . . . 7 ⊢ 𝐹:V–1-1-onto→V
6		permmodel.2	. . . . . . 7 ⊢ 𝑅 = (◡𝐹 ∘ E )
7		vex 3459	. . . . . . 7 ⊢ 𝑧 ∈ V
8		vex 3459	. . . . . . 7 ⊢ 𝑤 ∈ V
9	5, 6, 7, 8	brpermmodel 45712	. . . . . 6 ⊢ (𝑧𝑅𝑤 ↔ 𝑧 ∈ (𝐹‘𝑤))
10		vex 3459	. . . . . . 7 ⊢ 𝑥 ∈ V
11	5, 6, 8, 10	brpermmodel 45712	. . . . . 6 ⊢ (𝑤𝑅𝑥 ↔ 𝑤 ∈ (𝐹‘𝑥))
12		f1ofn 6821	. . . . . . . . 9 ⊢ (𝐹:V–1-1-onto→V → 𝐹 Fn V)
13	5, 12	ax-mp 5	. . . . . . . 8 ⊢ 𝐹 Fn V
14		ssv 3961	. . . . . . . 8 ⊢ (𝐹‘𝑥) ⊆ V
15		fnfvima 7231	. . . . . . . 8 ⊢ ((𝐹 Fn V ∧ (𝐹‘𝑥) ⊆ V ∧ 𝑤 ∈ (𝐹‘𝑥)) → (𝐹‘𝑤) ∈ (𝐹 “ (𝐹‘𝑥)))
16	13, 14, 15	mp3an12 1480	. . . . . . 7 ⊢ (𝑤 ∈ (𝐹‘𝑥) → (𝐹‘𝑤) ∈ (𝐹 “ (𝐹‘𝑥)))
17		elunii 4877	. . . . . . 7 ⊢ ((𝑧 ∈ (𝐹‘𝑤) ∧ (𝐹‘𝑤) ∈ (𝐹 “ (𝐹‘𝑥))) → 𝑧 ∈ ∪ (𝐹 “ (𝐹‘𝑥)))
18	16, 17	sylan2 604	. . . . . 6 ⊢ ((𝑧 ∈ (𝐹‘𝑤) ∧ 𝑤 ∈ (𝐹‘𝑥)) → 𝑧 ∈ ∪ (𝐹 “ (𝐹‘𝑥)))
19	9, 11, 18	syl2anb 609	. . . . 5 ⊢ ((𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧 ∈ ∪ (𝐹 “ (𝐹‘𝑥)))
20		f1ofun 6822	. . . . . . . . 9 ⊢ (𝐹:V–1-1-onto→V → Fun 𝐹)
21	5, 20	ax-mp 5	. . . . . . . 8 ⊢ Fun 𝐹
22		fvex 6894	. . . . . . . . 9 ⊢ (𝐹‘𝑥) ∈ V
23	22	funimaex 6623	. . . . . . . 8 ⊢ (Fun 𝐹 → (𝐹 “ (𝐹‘𝑥)) ∈ V)
24	21, 23	ax-mp 5	. . . . . . 7 ⊢ (𝐹 “ (𝐹‘𝑥)) ∈ V
25	24	uniex 7739	. . . . . 6 ⊢ ∪ (𝐹 “ (𝐹‘𝑥)) ∈ V
26	5, 6, 7, 25	brpermmodelcnv 45713	. . . . 5 ⊢ (𝑧𝑅(◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))) ↔ 𝑧 ∈ ∪ (𝐹 “ (𝐹‘𝑥)))
27	19, 26	sylibr 237	. . . 4 ⊢ ((𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧𝑅(◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))))
28	27	exlimiv 1960	. . 3 ⊢ (∃𝑤(𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧𝑅(◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))))
29	28	ax-gen 1825	. 2 ⊢ ∀𝑧(∃𝑤(𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧𝑅(◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))))
30	1, 4, 29	ceqsexv2d 3504	1 ⊢ ∃𝑦∀𝑧(∃𝑤(𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧𝑅𝑦)

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1568   = wceq 1570  ∃wex 1809   ∈ wcel 2143  Vcvv 3455   ⊆ wss 3905  ∪ cuni 4872   class class class wbr 5109   E cep 5560  ^◡ccnv 5660   “ cima 5664   ∘ ccom 5665  Fun wfun 6530   Fn wfn 6531  –1-1-onto→wf1o 6535  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-eprel 5561  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544

This theorem is referenced by: (None)