Theorem modelaxreplem3 44932

Description: Lemma for modelaxrep 44933. 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 44931	. 2 ⊢ (𝜓 → ran 𝐹 ∈ 𝑀)
11	1	sseld 3955	. . . . . . . . . 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 2222	. . . . . 6 ⊢ (𝜓 → (∃𝑤(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)) ↔ ∃𝑤(𝑧 ∈ 𝑀 ∧ (𝑤 ∈ 𝑀 ∧ (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))))
20	8	rneqi 5914	. . . . . . . 8 ⊢ ran 𝐹 = ran {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}
21		rnopab 5931	. . . . . . . 8 ⊢ ran {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))} = {𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}
22	20, 21	eqtri 2757	. . . . . . 7 ⊢ ran 𝐹 = {𝑧 ∣ ∃𝑤(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}
23	22	eqabri 2877	. . . . . 6 ⊢ (𝑧 ∈ ran 𝐹 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑)))
24		df-rex 3060	. . . . . . . 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 1952	. . . . . . 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 3239	. . 3 ⊢ (𝜓 → ∀𝑧 ∈ 𝑀 (𝑧 ∈ ran 𝐹 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
31	7	nfrn 5929	. . . . . 6 ⊢ Ⅎ𝑧ran 𝐹
32		sbcralt 3845	. . . . . 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 2914	. . . . . 6 ⊢ Ⅎ𝑧ran 𝐹 ∈ 𝑀
35		sbcbig 3815	. . . . . . 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 3830	. . . . . . . 8 ⊢ (ran 𝐹 ∈ 𝑀 → ([ran 𝐹 / 𝑦]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ran 𝐹))
37		nfcv 2897	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝑀
38		nfv 1913	. . . . . . . . . . 11 ⊢ Ⅎ𝑦 𝑤 ∈ 𝑥
39		nfa1 2150	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∀𝑦𝜑
40	38, 39	nfan 1898	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)
41	37, 40	nfrexw 3291	. . . . . . . . 9 ⊢ Ⅎ𝑦∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)
42	41	sbcgf 3834	. . . . . . . 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 3253	. . . . 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 44889	1 ⊢ (𝜓 → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1537   = wceq 1539  ∃wex 1778  Ⅎwnf 1782   ∈ wcel 2107  {cab 2712  Ⅎwnfc 2882  ∀wral 3050  ∃wrex 3059  [wsbc 3763   ⊆ wss 3924  ∅c0 4306  {copab 5178  dom cdm 5651  ran crn 5652  Fun wfun 6521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5246  ax-sep 5263  ax-nul 5273  ax-pow 5332  ax-pr 5399  ax-un 7723

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3357  df-reu 3358  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4881  df-iun 4966  df-br 5117  df-opab 5179  df-mpt 5199  df-id 5545  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6480  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-en 8954  df-dom 8955  df-sdom 8956

This theorem is referenced by:  modelaxrep  44933