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Theorem modelaxreplem3 44944
Description: Lemma for modelaxrep 44945. 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 (𝜓𝑥𝑀)
modelaxreplem.2 (𝜓 → ∀𝑓((Fun 𝑓 ∧ dom 𝑓𝑀 ∧ ran 𝑓𝑀) → ran 𝑓𝑀))
modelaxreplem.3 (𝜓 → ∅ ∈ 𝑀)
modelaxreplem.4 (𝜓𝑥𝑀)
modelaxreplem2.5 𝑤𝜓
modelaxreplem2.6 𝑧𝜓
modelaxreplem2.7 𝑧𝐹
modelaxreplem2.8 𝐹 = {⟨𝑤, 𝑧⟩ ∣ (𝑤𝑥 ∧ (𝑧𝑀 ∧ ∀𝑦𝜑))}
modelaxreplem2.9 (𝜓 → (𝑤𝑀 → ∃𝑦𝑀𝑧𝑀 (∀𝑦𝜑𝑧 = 𝑦)))
Assertion
Ref Expression
modelaxreplem3 (𝜓 → ∃𝑦𝑀𝑧𝑀 (𝑧𝑦 ↔ ∃𝑤𝑀 (𝑤𝑥 ∧ ∀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
StepHypRef Expression
1 modelaxreplem.1 . . 3 (𝜓𝑥𝑀)
2 modelaxreplem.2 . . 3 (𝜓 → ∀𝑓((Fun 𝑓 ∧ dom 𝑓𝑀 ∧ ran 𝑓𝑀) → ran 𝑓𝑀))
3 modelaxreplem.3 . . 3 (𝜓 → ∅ ∈ 𝑀)
4 modelaxreplem.4 . . 3 (𝜓𝑥𝑀)
5 modelaxreplem2.5 . . 3 𝑤𝜓
6 modelaxreplem2.6 . . 3 𝑧𝜓
7 modelaxreplem2.7 . . 3 𝑧𝐹
8 modelaxreplem2.8 . . 3 𝐹 = {⟨𝑤, 𝑧⟩ ∣ (𝑤𝑥 ∧ (𝑧𝑀 ∧ ∀𝑦𝜑))}
9 modelaxreplem2.9 . . 3 (𝜓 → (𝑤𝑀 → ∃𝑦𝑀𝑧𝑀 (∀𝑦𝜑𝑧 = 𝑦)))
101, 2, 3, 4, 5, 6, 7, 8, 9modelaxreplem2 44943 . 2 (𝜓 → ran 𝐹𝑀)
111sseld 3993 . . . . . . . . . 10 (𝜓 → (𝑤𝑥𝑤𝑀))
1211pm4.71rd 562 . . . . . . . . 9 (𝜓 → (𝑤𝑥 ↔ (𝑤𝑀𝑤𝑥)))
1312anbi1d 631 . . . . . . . 8 (𝜓 → ((𝑤𝑥 ∧ (𝑧𝑀 ∧ ∀𝑦𝜑)) ↔ ((𝑤𝑀𝑤𝑥) ∧ (𝑧𝑀 ∧ ∀𝑦𝜑))))
14 an12 645 . . . . . . . . 9 (((𝑤𝑀𝑤𝑥) ∧ (𝑧𝑀 ∧ ∀𝑦𝜑)) ↔ (𝑧𝑀 ∧ ((𝑤𝑀𝑤𝑥) ∧ ∀𝑦𝜑)))
15 anass 468 . . . . . . . . . 10 (((𝑤𝑀𝑤𝑥) ∧ ∀𝑦𝜑) ↔ (𝑤𝑀 ∧ (𝑤𝑥 ∧ ∀𝑦𝜑)))
1615anbi2i 623 . . . . . . . . 9 ((𝑧𝑀 ∧ ((𝑤𝑀𝑤𝑥) ∧ ∀𝑦𝜑)) ↔ (𝑧𝑀 ∧ (𝑤𝑀 ∧ (𝑤𝑥 ∧ ∀𝑦𝜑))))
1714, 16bitri 275 . . . . . . . 8 (((𝑤𝑀𝑤𝑥) ∧ (𝑧𝑀 ∧ ∀𝑦𝜑)) ↔ (𝑧𝑀 ∧ (𝑤𝑀 ∧ (𝑤𝑥 ∧ ∀𝑦𝜑))))
1813, 17bitrdi 287 . . . . . . 7 (𝜓 → ((𝑤𝑥 ∧ (𝑧𝑀 ∧ ∀𝑦𝜑)) ↔ (𝑧𝑀 ∧ (𝑤𝑀 ∧ (𝑤𝑥 ∧ ∀𝑦𝜑)))))
195, 18exbid 2220 . . . . . 6 (𝜓 → (∃𝑤(𝑤𝑥 ∧ (𝑧𝑀 ∧ ∀𝑦𝜑)) ↔ ∃𝑤(𝑧𝑀 ∧ (𝑤𝑀 ∧ (𝑤𝑥 ∧ ∀𝑦𝜑)))))
208rneqi 5950 . . . . . . . 8 ran 𝐹 = ran {⟨𝑤, 𝑧⟩ ∣ (𝑤𝑥 ∧ (𝑧𝑀 ∧ ∀𝑦𝜑))}
21 rnopab 5967 . . . . . . . 8 ran {⟨𝑤, 𝑧⟩ ∣ (𝑤𝑥 ∧ (𝑧𝑀 ∧ ∀𝑦𝜑))} = {𝑧 ∣ ∃𝑤(𝑤𝑥 ∧ (𝑧𝑀 ∧ ∀𝑦𝜑))}
2220, 21eqtri 2762 . . . . . . 7 ran 𝐹 = {𝑧 ∣ ∃𝑤(𝑤𝑥 ∧ (𝑧𝑀 ∧ ∀𝑦𝜑))}
2322eqabri 2882 . . . . . 6 (𝑧 ∈ ran 𝐹 ↔ ∃𝑤(𝑤𝑥 ∧ (𝑧𝑀 ∧ ∀𝑦𝜑)))
24 df-rex 3068 . . . . . . . 8 (∃𝑤𝑀 (𝑤𝑥 ∧ ∀𝑦𝜑) ↔ ∃𝑤(𝑤𝑀 ∧ (𝑤𝑥 ∧ ∀𝑦𝜑)))
2524anbi2i 623 . . . . . . 7 ((𝑧𝑀 ∧ ∃𝑤𝑀 (𝑤𝑥 ∧ ∀𝑦𝜑)) ↔ (𝑧𝑀 ∧ ∃𝑤(𝑤𝑀 ∧ (𝑤𝑥 ∧ ∀𝑦𝜑))))
26 19.42v 1950 . . . . . . 7 (∃𝑤(𝑧𝑀 ∧ (𝑤𝑀 ∧ (𝑤𝑥 ∧ ∀𝑦𝜑))) ↔ (𝑧𝑀 ∧ ∃𝑤(𝑤𝑀 ∧ (𝑤𝑥 ∧ ∀𝑦𝜑))))
2725, 26bitr4i 278 . . . . . 6 ((𝑧𝑀 ∧ ∃𝑤𝑀 (𝑤𝑥 ∧ ∀𝑦𝜑)) ↔ ∃𝑤(𝑧𝑀 ∧ (𝑤𝑀 ∧ (𝑤𝑥 ∧ ∀𝑦𝜑))))
2819, 23, 273bitr4g 314 . . . . 5 (𝜓 → (𝑧 ∈ ran 𝐹 ↔ (𝑧𝑀 ∧ ∃𝑤𝑀 (𝑤𝑥 ∧ ∀𝑦𝜑))))
2928baibd 539 . . . 4 ((𝜓𝑧𝑀) → (𝑧 ∈ ran 𝐹 ↔ ∃𝑤𝑀 (𝑤𝑥 ∧ ∀𝑦𝜑)))
306, 29ralrimia 3255 . . 3 (𝜓 → ∀𝑧𝑀 (𝑧 ∈ ran 𝐹 ↔ ∃𝑤𝑀 (𝑤𝑥 ∧ ∀𝑦𝜑)))
317nfrn 5965 . . . . . 6 𝑧ran 𝐹
32 sbcralt 3880 . . . . . 6 ((ran 𝐹𝑀𝑧ran 𝐹) → ([ran 𝐹 / 𝑦]𝑧𝑀 (𝑧𝑦 ↔ ∃𝑤𝑀 (𝑤𝑥 ∧ ∀𝑦𝜑)) ↔ ∀𝑧𝑀 [ran 𝐹 / 𝑦](𝑧𝑦 ↔ ∃𝑤𝑀 (𝑤𝑥 ∧ ∀𝑦𝜑))))
3331, 32mpan2 691 . . . . 5 (ran 𝐹𝑀 → ([ran 𝐹 / 𝑦]𝑧𝑀 (𝑧𝑦 ↔ ∃𝑤𝑀 (𝑤𝑥 ∧ ∀𝑦𝜑)) ↔ ∀𝑧𝑀 [ran 𝐹 / 𝑦](𝑧𝑦 ↔ ∃𝑤𝑀 (𝑤𝑥 ∧ ∀𝑦𝜑))))
3431nfel1 2919 . . . . . 6 𝑧ran 𝐹𝑀
35 sbcbig 3845 . . . . . . 7 (ran 𝐹𝑀 → ([ran 𝐹 / 𝑦](𝑧𝑦 ↔ ∃𝑤𝑀 (𝑤𝑥 ∧ ∀𝑦𝜑)) ↔ ([ran 𝐹 / 𝑦]𝑧𝑦[ran 𝐹 / 𝑦]𝑤𝑀 (𝑤𝑥 ∧ ∀𝑦𝜑))))
36 sbcel2gv 3862 . . . . . . . 8 (ran 𝐹𝑀 → ([ran 𝐹 / 𝑦]𝑧𝑦𝑧 ∈ ran 𝐹))
37 nfcv 2902 . . . . . . . . . 10 𝑦𝑀
38 nfv 1911 . . . . . . . . . . 11 𝑦 𝑤𝑥
39 nfa1 2148 . . . . . . . . . . 11 𝑦𝑦𝜑
4038, 39nfan 1896 . . . . . . . . . 10 𝑦(𝑤𝑥 ∧ ∀𝑦𝜑)
4137, 40nfrexw 3310 . . . . . . . . 9 𝑦𝑤𝑀 (𝑤𝑥 ∧ ∀𝑦𝜑)
4241sbcgf 3867 . . . . . . . 8 (ran 𝐹𝑀 → ([ran 𝐹 / 𝑦]𝑤𝑀 (𝑤𝑥 ∧ ∀𝑦𝜑) ↔ ∃𝑤𝑀 (𝑤𝑥 ∧ ∀𝑦𝜑)))
4336, 42bibi12d 345 . . . . . . 7 (ran 𝐹𝑀 → (([ran 𝐹 / 𝑦]𝑧𝑦[ran 𝐹 / 𝑦]𝑤𝑀 (𝑤𝑥 ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ ran 𝐹 ↔ ∃𝑤𝑀 (𝑤𝑥 ∧ ∀𝑦𝜑))))
4435, 43bitrd 279 . . . . . 6 (ran 𝐹𝑀 → ([ran 𝐹 / 𝑦](𝑧𝑦 ↔ ∃𝑤𝑀 (𝑤𝑥 ∧ ∀𝑦𝜑)) ↔ (𝑧 ∈ ran 𝐹 ↔ ∃𝑤𝑀 (𝑤𝑥 ∧ ∀𝑦𝜑))))
4534, 44ralbid 3270 . . . . 5 (ran 𝐹𝑀 → (∀𝑧𝑀 [ran 𝐹 / 𝑦](𝑧𝑦 ↔ ∃𝑤𝑀 (𝑤𝑥 ∧ ∀𝑦𝜑)) ↔ ∀𝑧𝑀 (𝑧 ∈ ran 𝐹 ↔ ∃𝑤𝑀 (𝑤𝑥 ∧ ∀𝑦𝜑))))
4633, 45bitrd 279 . . . 4 (ran 𝐹𝑀 → ([ran 𝐹 / 𝑦]𝑧𝑀 (𝑧𝑦 ↔ ∃𝑤𝑀 (𝑤𝑥 ∧ ∀𝑦𝜑)) ↔ ∀𝑧𝑀 (𝑧 ∈ ran 𝐹 ↔ ∃𝑤𝑀 (𝑤𝑥 ∧ ∀𝑦𝜑))))
4710, 46syl 17 . . 3 (𝜓 → ([ran 𝐹 / 𝑦]𝑧𝑀 (𝑧𝑦 ↔ ∃𝑤𝑀 (𝑤𝑥 ∧ ∀𝑦𝜑)) ↔ ∀𝑧𝑀 (𝑧 ∈ ran 𝐹 ↔ ∃𝑤𝑀 (𝑤𝑥 ∧ ∀𝑦𝜑))))
4830, 47mpbird 257 . 2 (𝜓[ran 𝐹 / 𝑦]𝑧𝑀 (𝑧𝑦 ↔ ∃𝑤𝑀 (𝑤𝑥 ∧ ∀𝑦𝜑)))
4910, 48rspesbcd 44935 1 (𝜓 → ∃𝑦𝑀𝑧𝑀 (𝑧𝑦 ↔ ∃𝑤𝑀 (𝑤𝑥 ∧ ∀𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086  wal 1534   = wceq 1536  wex 1775  wnf 1779  wcel 2105  {cab 2711  wnfc 2887  wral 3058  wrex 3067  [wsbc 3790  wss 3962  c0 4338  {copab 5209  dom cdm 5688  ran crn 5689  Fun wfun 6556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-en 8984  df-dom 8985  df-sdom 8986
This theorem is referenced by:  modelaxrep  44945
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