Theorem modelaxreplem3 46163

Description: Lemma for modelaxrep 46164. We show that the consequent of Replacement is satisfied with ran 𝐹 as the value of 𝑦. (Contributed by Eric Schmidt, 29-Sep-2025.)

Hypotheses

Ref	Expression
modelaxreplem.1	⊢ (𝜓 → 𝑥 ⊆ 𝑀)
modelaxreplem.2	⊢ (𝜓 → ∀𝑓((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) → ran 𝑓 ∈ 𝑀))
modelaxreplem.3	⊢ (𝜓 → ∅ ∈ 𝑀)
modelaxreplem.4	⊢ (𝜓 → 𝑥 ∈ 𝑀)
modelaxreplem2.5	⊢ Ⅎ𝑤𝜓
modelaxreplem2.6	⊢ Ⅎ𝑧𝜓
modelaxreplem2.7	⊢ Ⅎ𝑧𝐹
modelaxreplem2.8	⊢ 𝐹 = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}
modelaxreplem2.9	⊢ (𝜓 → (𝑤 ∈ 𝑀 → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)))

Assertion

Ref	Expression
modelaxreplem3	⊢ (𝜓 → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))

Distinct variable groups:   𝑦,𝑧,𝑤,𝑀   𝑓,𝐹   𝑓,𝑀   𝑥,𝑦,𝑧,𝑤

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧, 𝑤, 𝑓)   𝜓(𝑥, 𝑦, 𝑧, 𝑤, 𝑓)   𝐹(𝑥, 𝑦, 𝑧, 𝑤)   𝑀(𝑥)

Proof of Theorem modelaxreplem3

Step	Hyp	Ref	Expression
1		modelaxreplem.1	. . 3 ⊢ (𝜓 → 𝑥 ⊆ 𝑀)
2		modelaxreplem.2	. . 3 ⊢ (𝜓 → ∀𝑓((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) → ran 𝑓 ∈ 𝑀))
3		modelaxreplem.3	. . 3 ⊢ (𝜓 → ∅ ∈ 𝑀)
4		modelaxreplem.4	. . 3 ⊢ (𝜓 → 𝑥 ∈ 𝑀)
5		modelaxreplem2.5	. . 3 ⊢ Ⅎ𝑤𝜓
6		modelaxreplem2.6	. . 3 ⊢ Ⅎ𝑧𝜓
7		modelaxreplem2.7	. . 3 ⊢ Ⅎ𝑧𝐹
8		modelaxreplem2.8	. . 3 ⊢ 𝐹 = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}
9		modelaxreplem2.9	. . 3 ⊢ (𝜓 → (𝑤 ∈ 𝑀 → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)))
10	1, 2, 3, 4, 5, 6, 7, 8, 9	modelaxreplem2 46162	. 2 ⊢ (𝜓 → ran 𝐹 ∈ 𝑀)
11	1	sseld 3912	. . . . . . . . . 10 ⊢ (𝜓 → (𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑀))
12	11	pm4.71rd 572	. . . . . . . . 9 ⊢ (𝜓 → (𝑤 ∈ 𝑥 ↔ (𝑤 ∈ 𝑀 ∧ 𝑤 ∈ 𝑥)))
13	12	anbi1d 643	. . . . . . . 8 ⊢ (𝜓 → ((𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)) ↔ ((𝑤 ∈ 𝑀 ∧ 𝑤 ∈ 𝑥) ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))))
14		an12 658	. . . . . . . . 9 ⊢ (((𝑤 ∈ 𝑀 ∧ 𝑤 ∈ 𝑥) ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ 𝑀 ∧ ((𝑤 ∈ 𝑀 ∧ 𝑤 ∈ 𝑥) ∧ ∀𝑦𝜑)))
15		anass 474	. . . . . . . . . 10 ⊢ (((𝑤 ∈ 𝑀 ∧ 𝑤 ∈ 𝑥) ∧ ∀𝑦𝜑) ↔ (𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
16	15	anbi2i 635	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑀 ∧ ((𝑤 ∈ 𝑀 ∧ 𝑤 ∈ 𝑥) ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
17	14, 16	bitri 278	. . . . . . . 8 ⊢ (((𝑤 ∈ 𝑀 ∧ 𝑤 ∈ 𝑥) ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
18	13, 17	bitrdi 290	. . . . . . 7 ⊢ (𝜓 → ((𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))))
19	5, 18	exbid 2263	. . . . . 6 ⊢ (𝜓 → (∃𝑤(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)) ↔ ∃𝑤(𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))))
20	8	rneqi 5925	. . . . . . . 8 ⊢ ran 𝐹 = ran {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}
21		rnopab 5942	. . . . . . . 8 ⊢ ran {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))} = {𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}
22	20, 21	eqtri 2786	. . . . . . 7 ⊢ ran 𝐹 = {𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}
23	22	eqabri 2906	. . . . . 6 ⊢ (𝑧 ∈ ran 𝐹 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)))
24		df-rex 3091	. . . . . . . 8 ⊢ (∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑) ↔ ∃𝑤(𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
25	24	anbi2i 635	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑀 ∧ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ 𝑀 ∧ ∃𝑤(𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
26		19.42v 1988	. . . . . . 7 ⊢ (∃𝑤(𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))) ↔ (𝑧 ∈ 𝑀 ∧ ∃𝑤(𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
27	25, 26	bitr4i 281	. . . . . 6 ⊢ ((𝑧 ∈ 𝑀 ∧ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)) ↔ ∃𝑤(𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
28	19, 23, 27	3bitr4g 317	. . . . 5 ⊢ (𝜓 → (𝑧 ∈ ran 𝐹 ↔ (𝑧 ∈ 𝑀 ∧ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
29	28	baibd 549	. . . 4 ⊢ ((𝜓 ∧ 𝑧 ∈ 𝑀) → (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
30	6, 29	ralrimia 3265	. . 3 ⊢ (𝜓 → ∀𝑧 ∈ 𝑀 (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
31	7	nfrn 5940	. . . . . 6 ⊢ Ⅎ𝑧ran 𝐹
32		sbcralt 3823	. . . . . 6 ⊢ ((ran 𝐹 ∈ 𝑀 ∧ Ⅎ𝑧ran 𝐹) → ([ran 𝐹 / 𝑦]∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)) ↔ ∀𝑧 ∈ 𝑀 [ran 𝐹 / 𝑦](𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
33	31, 32	mpan2 704	. . . . 5 ⊢ (ran 𝐹 ∈ 𝑀 → ([ran 𝐹 / 𝑦]∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)) ↔ ∀𝑧 ∈ 𝑀 [ran 𝐹 / 𝑦](𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
34	31	nfel1 2942	. . . . . 6 ⊢ Ⅎ𝑧ran 𝐹 ∈ 𝑀
35		sbcbig 3793	. . . . . . 7 ⊢ (ran 𝐹 ∈ 𝑀 → ([ran 𝐹 / 𝑦](𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)) ↔ ([ran 𝐹 / 𝑦]𝑧 ∈ 𝑦 ↔ [ran 𝐹 / 𝑦]∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
36		sbcel2gv 3808	. . . . . . . 8 ⊢ (ran 𝐹 ∈ 𝑀 → ([ran 𝐹 / 𝑦]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ran 𝐹))
37		nfcv 2926	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝑀
38		nfv 1949	. . . . . . . . . . 11 ⊢ Ⅎ𝑦 𝑤 ∈ 𝑥
39		nfa1 2191	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∀𝑦𝜑
40	38, 39	nfan 1934	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)
41	37, 40	nfrexw 3314	. . . . . . . . 9 ⊢ Ⅎ𝑦∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)
42	41	sbcgf 3812	. . . . . . . 8 ⊢ (ran 𝐹 ∈ 𝑀 → ([ran 𝐹 / 𝑦]∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑) ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
43	36, 42	bibi12d 348	. . . . . . 7 ⊢ (ran 𝐹 ∈ 𝑀 → (([ran 𝐹 / 𝑦]𝑧 ∈ 𝑦 ↔ [ran 𝐹 / 𝑦]∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
44	35, 43	bitrd 282	. . . . . 6 ⊢ (ran 𝐹 ∈ 𝑀 → ([ran 𝐹 / 𝑦](𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
45	34, 44	ralbid 3279	. . . . 5 ⊢ (ran 𝐹 ∈ 𝑀 → (∀𝑧 ∈ 𝑀 [ran 𝐹 / 𝑦](𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)) ↔ ∀𝑧 ∈ 𝑀 (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
46	33, 45	bitrd 282	. . . 4 ⊢ (ran 𝐹 ∈ 𝑀 → ([ran 𝐹 / 𝑦]∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)) ↔ ∀𝑧 ∈ 𝑀 (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
47	10, 46	syl 18	. . 3 ⊢ (𝜓 → ([ran 𝐹 / 𝑦]∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)) ↔ ∀𝑧 ∈ 𝑀 (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
48	30, 47	mpbird 260	. 2 ⊢ (𝜓 → [ran 𝐹 / 𝑦]∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
49	10, 48	rspesbcd 46109	1 ⊢ (𝜓 → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1812  Ⅎwnf 1816   ∈ wcel 2147  {cab 2741  Ⅎwnfc 2911  ∀wral 3080  ∃wrex 3090  [wsbc 3742   ⊆ wss 3900  ∅c0 4281  {copab 5170  dom cdm 5655  ran crn 5656  Fun wfun 6549

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-en 8999  df-dom 9000  df-sdom 9001

This theorem is used by:  modelaxrep  46164