Theorem permaxun 45367

Description: The Axiom of Union ax-un 7690 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 6855	. 2 ⊢ (◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))) ∈ V
2		breq2 5104	. . . 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 1922	. 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 3446	. . . . . . 7 ⊢ 𝑧 ∈ V
8		vex 3446	. . . . . . 7 ⊢ 𝑤 ∈ V
9	5, 6, 7, 8	brpermmodel 45359	. . . . . 6 ⊢ (𝑧𝑅𝑤 ↔ 𝑧 ∈ (𝐹‘𝑤))
10		vex 3446	. . . . . . 7 ⊢ 𝑥 ∈ V
11	5, 6, 8, 10	brpermmodel 45359	. . . . . 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 3960	. . . . . . . 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 1454	. . . . . . 7 ⊢ (𝑤 ∈ (𝐹‘𝑥) → (𝐹‘𝑤) ∈ (𝐹 “ (𝐹‘𝑥)))
17		elunii 4870	. . . . . . 7 ⊢ ((𝑧 ∈ (𝐹‘𝑤) ∧ (𝐹‘𝑤) ∈ (𝐹 “ (𝐹‘𝑥))) → 𝑧 ∈ ∪ (𝐹 “ (𝐹‘𝑥)))
18	16, 17	sylan2 594	. . . . . 6 ⊢ ((𝑧 ∈ (𝐹‘𝑤) ∧ 𝑤 ∈ (𝐹‘𝑥)) → 𝑧 ∈ ∪ (𝐹 “ (𝐹‘𝑥)))
19	9, 11, 18	syl2anb 599	. . . . 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 6855	. . . . . . . . 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 7696	. . . . . 6 ⊢ ∪ (𝐹 “ (𝐹‘𝑥)) ∈ V
26	5, 6, 7, 25	brpermmodelcnv 45360	. . . . 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 1932	. . 3 ⊢ (∃𝑤(𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧𝑅(◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))))
29	28	ax-gen 1797	. 2 ⊢ ∀𝑧(∃𝑤(𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧𝑅(◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))))
30	1, 4, 29	ceqsexv2d 3493	1 ⊢ ∃𝑦∀𝑧(∃𝑤(𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧𝑅𝑦)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3442   ⊆ wss 3903  ∪ cuni 4865   class class class wbr 5100   E cep 5531  ^◡ccnv 5631   “ cima 5635   ∘ ccom 5636  Fun wfun 6494   Fn wfn 6495  –1-1-onto→wf1o 6499  ‘cfv 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-eprel 5532  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508

This theorem is referenced by: (None)