Theorem permaxun 44994

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 6873	. 2 ⊢ (◡𝐹‘∪ (𝐹 “ (𝐹‘𝑥))) ∈ V
2		breq2 5113	. . . 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 3454	. . . . . . 7 ⊢ 𝑧 ∈ V
8		vex 3454	. . . . . . 7 ⊢ 𝑤 ∈ V
9	5, 6, 7, 8	brpermmodel 44986	. . . . . 6 ⊢ (𝑧𝑅𝑤 ↔ 𝑧 ∈ (𝐹‘𝑤))
10		vex 3454	. . . . . . 7 ⊢ 𝑥 ∈ V
11	5, 6, 8, 10	brpermmodel 44986	. . . . . 6 ⊢ (𝑤𝑅𝑥 ↔ 𝑤 ∈ (𝐹‘𝑥))
12		f1ofn 6803	. . . . . . . . 9 ⊢ (𝐹:V–1-1-onto→V → 𝐹 Fn V)
13	5, 12	ax-mp 5	. . . . . . . 8 ⊢ 𝐹 Fn V
14		ssv 3973	. . . . . . . 8 ⊢ (𝐹‘𝑥) ⊆ V
15		fnfvima 7209	. . . . . . . 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 4878	. . . . . . 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 6804	. . . . . . . . 9 ⊢ (𝐹:V–1-1-onto→V → Fun 𝐹)
21	5, 20	ax-mp 5	. . . . . . . 8 ⊢ Fun 𝐹
22		fvex 6873	. . . . . . . . 9 ⊢ (𝐹‘𝑥) ∈ V
23	22	funimaex 6607	. . . . . . . 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 44987	. . . . 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 3502	1 ⊢ ∃𝑦∀𝑧(∃𝑤(𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧𝑅𝑦)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3450   ⊆ wss 3916  ∪ cuni 4873   class class class wbr 5109   E cep 5539  ^◡ccnv 5639   “ cima 5643   ∘ ccom 5644  Fun wfun 6507   Fn wfn 6508  –1-1-onto→wf1o 6512  ‘cfv 6513

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 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pr 5389  ax-un 7713

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-id 5535  df-eprel 5540  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521

This theorem is referenced by: (None)