Theorem modelaxreplem3 44964

Description: Lemma for modelaxrep 44965. 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 44963	. 2 ⊢ (𝜓 → ran 𝐹 ∈ 𝑀)
11	1	sseld 3934	. . . . . . . . . 10 ⊢ (𝜓 → (𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑀))
12	11	pm4.71rd 562	. . . . . . . . 9 ⊢ (𝜓 → (𝑤 ∈ 𝑥 ↔ (𝑤 ∈ 𝑀 ∧ 𝑤 ∈ 𝑥)))
13	12	anbi1d 631	. . . . . . . 8 ⊢ (𝜓 → ((𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)) ↔ ((𝑤 ∈ 𝑀 ∧ 𝑤 ∈ 𝑥) ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))))
14		an12 645	. . . . . . . . 9 ⊢ (((𝑤 ∈ 𝑀 ∧ 𝑤 ∈ 𝑥) ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ 𝑀 ∧ ((𝑤 ∈ 𝑀 ∧ 𝑤 ∈ 𝑥) ∧ ∀𝑦𝜑)))
15		anass 468	. . . . . . . . . 10 ⊢ (((𝑤 ∈ 𝑀 ∧ 𝑤 ∈ 𝑥) ∧ ∀𝑦𝜑) ↔ (𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
16	15	anbi2i 623	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑀 ∧ ((𝑤 ∈ 𝑀 ∧ 𝑤 ∈ 𝑥) ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
17	14, 16	bitri 275	. . . . . . . 8 ⊢ (((𝑤 ∈ 𝑀 ∧ 𝑤 ∈ 𝑥) ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
18	13, 17	bitrdi 287	. . . . . . 7 ⊢ (𝜓 → ((𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))))
19	5, 18	exbid 2224	. . . . . 6 ⊢ (𝜓 → (∃𝑤(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)) ↔ ∃𝑤(𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))))
20	8	rneqi 5879	. . . . . . . 8 ⊢ ran 𝐹 = ran {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}
21		rnopab 5896	. . . . . . . 8 ⊢ ran {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))} = {𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}
22	20, 21	eqtri 2752	. . . . . . 7 ⊢ ran 𝐹 = {𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}
23	22	eqabri 2871	. . . . . 6 ⊢ (𝑧 ∈ ran 𝐹 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)))
24		df-rex 3054	. . . . . . . 8 ⊢ (∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑) ↔ ∃𝑤(𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
25	24	anbi2i 623	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑀 ∧ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ 𝑀 ∧ ∃𝑤(𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
26		19.42v 1953	. . . . . . 7 ⊢ (∃𝑤(𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))) ↔ (𝑧 ∈ 𝑀 ∧ ∃𝑤(𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
27	25, 26	bitr4i 278	. . . . . 6 ⊢ ((𝑧 ∈ 𝑀 ∧ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)) ↔ ∃𝑤(𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
28	19, 23, 27	3bitr4g 314	. . . . 5 ⊢ (𝜓 → (𝑧 ∈ ran 𝐹 ↔ (𝑧 ∈ 𝑀 ∧ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
29	28	baibd 539	. . . 4 ⊢ ((𝜓 ∧ 𝑧 ∈ 𝑀) → (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
30	6, 29	ralrimia 3228	. . 3 ⊢ (𝜓 → ∀𝑧 ∈ 𝑀 (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
31	7	nfrn 5894	. . . . . 6 ⊢ Ⅎ𝑧ran 𝐹
32		sbcralt 3824	. . . . . 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 691	. . . . 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 2908	. . . . . 6 ⊢ Ⅎ𝑧ran 𝐹 ∈ 𝑀
35		sbcbig 3794	. . . . . . 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 3809	. . . . . . . 8 ⊢ (ran 𝐹 ∈ 𝑀 → ([ran 𝐹 / 𝑦]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ran 𝐹))
37		nfcv 2891	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝑀
38		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑦 𝑤 ∈ 𝑥
39		nfa1 2152	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∀𝑦𝜑
40	38, 39	nfan 1899	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)
41	37, 40	nfrexw 3277	. . . . . . . . 9 ⊢ Ⅎ𝑦∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)
42	41	sbcgf 3813	. . . . . . . 8 ⊢ (ran 𝐹 ∈ 𝑀 → ([ran 𝐹 / 𝑦]∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑) ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
43	36, 42	bibi12d 345	. . . . . . 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 279	. . . . . 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 3242	. . . . 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 279	. . . 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 17	. . 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 257	. 2 ⊢ (𝜓 → [ran 𝐹 / 𝑦]∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
49	10, 48	rspesbcd 44921	1 ⊢ (𝜓 → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540  ∃wex 1779  Ⅎwnf 1783   ∈ wcel 2109  {cab 2707  Ⅎwnfc 2876  ∀wral 3044  ∃wrex 3053  [wsbc 3742   ⊆ wss 3903  ∅c0 4284  {copab 5154  dom cdm 5619  ran crn 5620  Fun wfun 6476

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 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

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-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-en 8873  df-dom 8874  df-sdom 8875

This theorem is referenced by:  modelaxrep  44965