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Theorem modelaxreplem2 44987
Description: Lemma for modelaxrep 44989. We define a class 𝐹 and show that the antecedent of Replacement implies that 𝐹 is a function. We use Replacement (in the form of funex 7237) to show that 𝐹 exists. Then we show that, under our hypotheses, the range of 𝐹 is a member of 𝑀. (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
modelaxreplem2 (𝜓 → ran 𝐹𝑀)
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 modelaxreplem2
StepHypRef Expression
1 modelaxreplem2.5 . . . 4 𝑤𝜓
2 modelaxreplem.1 . . . . . . 7 (𝜓𝑥𝑀)
32sseld 3981 . . . . . 6 (𝜓 → (𝑤𝑥𝑤𝑀))
4 modelaxreplem2.9 . . . . . . 7 (𝜓 → (𝑤𝑀 → ∃𝑦𝑀𝑧𝑀 (∀𝑦𝜑𝑧 = 𝑦)))
5 nfa1 2151 . . . . . . . . 9 𝑦𝑦𝜑
65rmo2i 3887 . . . . . . . 8 (∃𝑦𝑀𝑧𝑀 (∀𝑦𝜑𝑧 = 𝑦) → ∃*𝑧𝑀𝑦𝜑)
7 df-rmo 3379 . . . . . . . 8 (∃*𝑧𝑀𝑦𝜑 ↔ ∃*𝑧(𝑧𝑀 ∧ ∀𝑦𝜑))
86, 7sylib 218 . . . . . . 7 (∃𝑦𝑀𝑧𝑀 (∀𝑦𝜑𝑧 = 𝑦) → ∃*𝑧(𝑧𝑀 ∧ ∀𝑦𝜑))
94, 8syl6 35 . . . . . 6 (𝜓 → (𝑤𝑀 → ∃*𝑧(𝑧𝑀 ∧ ∀𝑦𝜑)))
103, 9syld 47 . . . . 5 (𝜓 → (𝑤𝑥 → ∃*𝑧(𝑧𝑀 ∧ ∀𝑦𝜑)))
11 moanimv 2618 . . . . 5 (∃*𝑧(𝑤𝑥 ∧ (𝑧𝑀 ∧ ∀𝑦𝜑)) ↔ (𝑤𝑥 → ∃*𝑧(𝑧𝑀 ∧ ∀𝑦𝜑)))
1210, 11sylibr 234 . . . 4 (𝜓 → ∃*𝑧(𝑤𝑥 ∧ (𝑧𝑀 ∧ ∀𝑦𝜑)))
131, 12alrimi 2213 . . 3 (𝜓 → ∀𝑤∃*𝑧(𝑤𝑥 ∧ (𝑧𝑀 ∧ ∀𝑦𝜑)))
14 modelaxreplem2.8 . . . . 5 𝐹 = {⟨𝑤, 𝑧⟩ ∣ (𝑤𝑥 ∧ (𝑧𝑀 ∧ ∀𝑦𝜑))}
1514funeqi 6585 . . . 4 (Fun 𝐹 ↔ Fun {⟨𝑤, 𝑧⟩ ∣ (𝑤𝑥 ∧ (𝑧𝑀 ∧ ∀𝑦𝜑))})
16 funopab 6599 . . . 4 (Fun {⟨𝑤, 𝑧⟩ ∣ (𝑤𝑥 ∧ (𝑧𝑀 ∧ ∀𝑦𝜑))} ↔ ∀𝑤∃*𝑧(𝑤𝑥 ∧ (𝑧𝑀 ∧ ∀𝑦𝜑)))
1715, 16bitri 275 . . 3 (Fun 𝐹 ↔ ∀𝑤∃*𝑧(𝑤𝑥 ∧ (𝑧𝑀 ∧ ∀𝑦𝜑)))
1813, 17sylibr 234 . 2 (𝜓 → Fun 𝐹)
19 modelaxreplem.2 . . 3 (𝜓 → ∀𝑓((Fun 𝑓 ∧ dom 𝑓𝑀 ∧ ran 𝑓𝑀) → ran 𝑓𝑀))
20 modelaxreplem.3 . . 3 (𝜓 → ∅ ∈ 𝑀)
21 modelaxreplem.4 . . 3 (𝜓𝑥𝑀)
2214dmeqi 5913 . . . 4 dom 𝐹 = dom {⟨𝑤, 𝑧⟩ ∣ (𝑤𝑥 ∧ (𝑧𝑀 ∧ ∀𝑦𝜑))}
23 dmopabss 5927 . . . 4 dom {⟨𝑤, 𝑧⟩ ∣ (𝑤𝑥 ∧ (𝑧𝑀 ∧ ∀𝑦𝜑))} ⊆ 𝑥
2422, 23eqsstri 4029 . . 3 dom 𝐹𝑥
252, 19, 20, 21, 24modelaxreplem1 44986 . 2 (𝜓 → dom 𝐹𝑀)
26 an12 645 . . . . . . 7 ((𝑤𝑥 ∧ (𝑧𝑀 ∧ ∀𝑦𝜑)) ↔ (𝑧𝑀 ∧ (𝑤𝑥 ∧ ∀𝑦𝜑)))
2726opabbii 5208 . . . . . 6 {⟨𝑤, 𝑧⟩ ∣ (𝑤𝑥 ∧ (𝑧𝑀 ∧ ∀𝑦𝜑))} = {⟨𝑤, 𝑧⟩ ∣ (𝑧𝑀 ∧ (𝑤𝑥 ∧ ∀𝑦𝜑))}
2814, 27eqtri 2764 . . . . 5 𝐹 = {⟨𝑤, 𝑧⟩ ∣ (𝑧𝑀 ∧ (𝑤𝑥 ∧ ∀𝑦𝜑))}
2928rneqi 5946 . . . 4 ran 𝐹 = ran {⟨𝑤, 𝑧⟩ ∣ (𝑧𝑀 ∧ (𝑤𝑥 ∧ ∀𝑦𝜑))}
30 rnopabss 5964 . . . 4 ran {⟨𝑤, 𝑧⟩ ∣ (𝑧𝑀 ∧ (𝑤𝑥 ∧ ∀𝑦𝜑))} ⊆ 𝑀
3129, 30eqsstri 4029 . . 3 ran 𝐹𝑀
3231a1i 11 . 2 (𝜓 → ran 𝐹𝑀)
33 funex 7237 . . . 4 ((Fun 𝐹 ∧ dom 𝐹𝑀) → 𝐹 ∈ V)
3418, 25, 33syl2anc 584 . . 3 (𝜓𝐹 ∈ V)
35 funeq 6584 . . . . . 6 (𝑓 = 𝐹 → (Fun 𝑓 ↔ Fun 𝐹))
36 dmeq 5912 . . . . . . 7 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
3736eleq1d 2825 . . . . . 6 (𝑓 = 𝐹 → (dom 𝑓𝑀 ↔ dom 𝐹𝑀))
38 rneq 5945 . . . . . . 7 (𝑓 = 𝐹 → ran 𝑓 = ran 𝐹)
3938sseq1d 4014 . . . . . 6 (𝑓 = 𝐹 → (ran 𝑓𝑀 ↔ ran 𝐹𝑀))
4035, 37, 393anbi123d 1438 . . . . 5 (𝑓 = 𝐹 → ((Fun 𝑓 ∧ dom 𝑓𝑀 ∧ ran 𝑓𝑀) ↔ (Fun 𝐹 ∧ dom 𝐹𝑀 ∧ ran 𝐹𝑀)))
4138eleq1d 2825 . . . . 5 (𝑓 = 𝐹 → (ran 𝑓𝑀 ↔ ran 𝐹𝑀))
4240, 41imbi12d 344 . . . 4 (𝑓 = 𝐹 → (((Fun 𝑓 ∧ dom 𝑓𝑀 ∧ ran 𝑓𝑀) → ran 𝑓𝑀) ↔ ((Fun 𝐹 ∧ dom 𝐹𝑀 ∧ ran 𝐹𝑀) → ran 𝐹𝑀)))
4342spcgv 3595 . . 3 (𝐹 ∈ V → (∀𝑓((Fun 𝑓 ∧ dom 𝑓𝑀 ∧ ran 𝑓𝑀) → ran 𝑓𝑀) → ((Fun 𝐹 ∧ dom 𝐹𝑀 ∧ ran 𝐹𝑀) → ran 𝐹𝑀)))
4434, 19, 43sylc 65 . 2 (𝜓 → ((Fun 𝐹 ∧ dom 𝐹𝑀 ∧ ran 𝐹𝑀) → ran 𝐹𝑀))
4518, 25, 32, 44mp3and 1466 1 (𝜓 → ran 𝐹𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087  wal 1538   = wceq 1540  wnf 1783  wcel 2108  ∃*wmo 2537  wnfc 2889  wral 3060  wrex 3069  ∃*wrmo 3378  Vcvv 3479  wss 3950  c0 4332  {copab 5203  dom cdm 5683  ran crn 5684  Fun wfun 6553
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5277  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-en 8982  df-dom 8983  df-sdom 8984
This theorem is referenced by:  modelaxreplem3  44988
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