Theorem permaxun 44974

Description: The Axiom of Union ax-un 7691 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 6853	. 2 ⊢ (◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))) ∈ V
2		breq2 5106	. . . 4 ⊢ (𝑦 = (◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))) → (𝑧𝑅𝑦 ↔ 𝑧𝑅(◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥)))))
3	2	imbi2d 340	. . 3 ⊢ (𝑦 = (◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))) → ((∃𝑤(𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧𝑅𝑦) ↔ (∃𝑤(𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧𝑅(◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))))))
4	3	albidv 1920	. 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 3448	. . . . . . 7 ⊢ 𝑧 ∈ V
8		vex 3448	. . . . . . 7 ⊢ 𝑤 ∈ V
9	5, 6, 7, 8	brpermmodel 44966	. . . . . 6 ⊢ (𝑧𝑅𝑤 ↔ 𝑧 ∈ (𝐹‘𝑤))
10		vex 3448	. . . . . . 7 ⊢ 𝑥 ∈ V
11	5, 6, 8, 10	brpermmodel 44966	. . . . . 6 ⊢ (𝑤𝑅𝑥 ↔ 𝑤 ∈ (𝐹‘𝑥))
12		f1ofn 6783	. . . . . . . . 9 ⊢ (𝐹:V–1-1-onto→V → 𝐹 Fn V)
13	5, 12	ax-mp 5	. . . . . . . 8 ⊢ 𝐹 Fn V
14		ssv 3968	. . . . . . . 8 ⊢ (𝐹‘𝑥) ⊆ V
15		fnfvima 7189	. . . . . . . 8 ⊢ ((𝐹 Fn V ∧ (𝐹‘𝑥) ⊆ V ∧ 𝑤 ∈ (𝐹‘𝑥)) → (𝐹‘𝑤) ∈ (𝐹 “ (𝐹‘𝑥)))
16	13, 14, 15	mp3an12 1453	. . . . . . 7 ⊢ (𝑤 ∈ (𝐹‘𝑥) → (𝐹‘𝑤) ∈ (𝐹 “ (𝐹‘𝑥)))
17		elunii 4872	. . . . . . 7 ⊢ ((𝑧 ∈ (𝐹‘𝑤) ∧ (𝐹‘𝑤) ∈ (𝐹 “ (𝐹‘𝑥))) → 𝑧 ∈ ∪ (𝐹 “ (𝐹‘𝑥)))
18	16, 17	sylan2 593	. . . . . 6 ⊢ ((𝑧 ∈ (𝐹‘𝑤) ∧ 𝑤 ∈ (𝐹‘𝑥)) → 𝑧 ∈ ∪ (𝐹 “ (𝐹‘𝑥)))
19	9, 11, 18	syl2anb 598	. . . . 5 ⊢ ((𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧 ∈ ∪ (𝐹 “ (𝐹‘𝑥)))
20		f1ofun 6784	. . . . . . . . 9 ⊢ (𝐹:V–1-1-onto→V → Fun 𝐹)
21	5, 20	ax-mp 5	. . . . . . . 8 ⊢ Fun 𝐹
22		fvex 6853	. . . . . . . . 9 ⊢ (𝐹‘𝑥) ∈ V
23	22	funimaex 6588	. . . . . . . 8 ⊢ (Fun 𝐹 → (𝐹 “ (𝐹‘𝑥)) ∈ V)
24	21, 23	ax-mp 5	. . . . . . 7 ⊢ (𝐹 “ (𝐹‘𝑥)) ∈ V
25	24	uniex 7697	. . . . . 6 ⊢ ∪ (𝐹 “ (𝐹‘𝑥)) ∈ V
26	5, 6, 7, 25	brpermmodelcnv 44967	. . . . 5 ⊢ (𝑧𝑅(◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))) ↔ 𝑧 ∈ ∪ (𝐹 “ (𝐹‘𝑥)))
27	19, 26	sylibr 234	. . . 4 ⊢ ((𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧𝑅(◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))))
28	27	exlimiv 1930	. . 3 ⊢ (∃𝑤(𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧𝑅(◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))))
29	28	ax-gen 1795	. 2 ⊢ ∀𝑧(∃𝑤(𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧𝑅(◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))))
30	1, 4, 29	ceqsexv2d 3496	1 ⊢ ∃𝑦∀𝑧(∃𝑤(𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧𝑅𝑦)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3444   ⊆ wss 3911  ∪ cuni 4867   class class class wbr 5102   E cep 5530  ^◡ccnv 5630   “ cima 5634   ∘ ccom 5635  Fun wfun 6493   Fn wfn 6494  –1-1-onto→wf1o 6498  ‘cfv 6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-eprel 5531  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507

This theorem is referenced by: (None)