Theorem permaxun 45548

Description: The Axiom of Union ax-un 7713 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 6875	. 2 ⊢ (◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))) ∈ V
2		breq2 5101	. . . 4 ⊢ (𝑦 = (◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))) → (𝑧𝑅𝑦 ↔ 𝑧𝑅(◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥)))))
3	2	imbi2d 342	. . 3 ⊢ (𝑦 = (◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))) → ((∃𝑤(𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧𝑅𝑦) ↔ (∃𝑤(𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧𝑅(◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))))))
4	3	albidv 1939	. 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 3457	. . . . . . 7 ⊢ 𝑧 ∈ V
8		vex 3457	. . . . . . 7 ⊢ 𝑤 ∈ V
9	5, 6, 7, 8	brpermmodel 45540	. . . . . 6 ⊢ (𝑧𝑅𝑤 ↔ 𝑧 ∈ (𝐹‘𝑤))
10		vex 3457	. . . . . . 7 ⊢ 𝑥 ∈ V
11	5, 6, 8, 10	brpermmodel 45540	. . . . . 6 ⊢ (𝑤𝑅𝑥 ↔ 𝑤 ∈ (𝐹‘𝑥))
12		f1ofn 6802	. . . . . . . . 9 ⊢ (𝐹:V–1-1-onto→V → 𝐹 Fn V)
13	5, 12	ax-mp 5	. . . . . . . 8 ⊢ 𝐹 Fn V
14		ssv 3958	. . . . . . . 8 ⊢ (𝐹‘𝑥) ⊆ V
15		fnfvima 7212	. . . . . . . 8 ⊢ ((𝐹 Fn V ∧ (𝐹‘𝑥) ⊆ V ∧ 𝑤 ∈ (𝐹‘𝑥)) → (𝐹‘𝑤) ∈ (𝐹 “ (𝐹‘𝑥)))
16	13, 14, 15	mp3an12 1471	. . . . . . 7 ⊢ (𝑤 ∈ (𝐹‘𝑥) → (𝐹‘𝑤) ∈ (𝐹 “ (𝐹‘𝑥)))
17		elunii 4867	. . . . . . 7 ⊢ ((𝑧 ∈ (𝐹‘𝑤) ∧ (𝐹‘𝑤) ∈ (𝐹 “ (𝐹‘𝑥))) → 𝑧 ∈ ∪ (𝐹 “ (𝐹‘𝑥)))
18	16, 17	sylan2 602	. . . . . 6 ⊢ ((𝑧 ∈ (𝐹‘𝑤) ∧ 𝑤 ∈ (𝐹‘𝑥)) → 𝑧 ∈ ∪ (𝐹 “ (𝐹‘𝑥)))
19	9, 11, 18	syl2anb 607	. . . . 5 ⊢ ((𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧 ∈ ∪ (𝐹 “ (𝐹‘𝑥)))
20		f1ofun 6803	. . . . . . . . 9 ⊢ (𝐹:V–1-1-onto→V → Fun 𝐹)
21	5, 20	ax-mp 5	. . . . . . . 8 ⊢ Fun 𝐹
22		fvex 6875	. . . . . . . . 9 ⊢ (𝐹‘𝑥) ∈ V
23	22	funimaex 6604	. . . . . . . 8 ⊢ (Fun 𝐹 → (𝐹 “ (𝐹‘𝑥)) ∈ V)
24	21, 23	ax-mp 5	. . . . . . 7 ⊢ (𝐹 “ (𝐹‘𝑥)) ∈ V
25	24	uniex 7719	. . . . . 6 ⊢ ∪ (𝐹 “ (𝐹‘𝑥)) ∈ V
26	5, 6, 7, 25	brpermmodelcnv 45541	. . . . 5 ⊢ (𝑧𝑅(◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))) ↔ 𝑧 ∈ ∪ (𝐹 “ (𝐹‘𝑥)))
27	19, 26	sylibr 236	. . . 4 ⊢ ((𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧𝑅(◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))))
28	27	exlimiv 1949	. . 3 ⊢ (∃𝑤(𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧𝑅(◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))))
29	28	ax-gen 1814	. 2 ⊢ ∀𝑧(∃𝑤(𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧𝑅(◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))))
30	1, 4, 29	ceqsexv2d 3502	1 ⊢ ∃𝑦∀𝑧(∃𝑤(𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧𝑅𝑦)

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1557   = wceq 1559  ∃wex 1798   ∈ wcel 2141  Vcvv 3453   ⊆ wss 3902  ∪ cuni 4862   class class class wbr 5097   E cep 5542  ^◡ccnv 5642   “ cima 5646   ∘ ccom 5647  Fun wfun 6510   Fn wfn 6511  –1-1-onto→wf1o 6515  ‘cfv 6516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-id 5538  df-eprel 5543  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524

This theorem is referenced by: (None)