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Theorem modelaxreplem1 44995
Description: Lemma for modelaxrep 44998. We show that 𝑀 is closed under taking subsets. (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)
modelaxreplem1.5 𝐴𝑥
Assertion
Ref Expression
modelaxreplem1 (𝜓𝐴𝑀)
Distinct variable group:   𝑓,𝑀
Allowed substitution hints:   𝜓(𝑥,𝑓)   𝐴(𝑥,𝑓)   𝑀(𝑥)

Proof of Theorem modelaxreplem1
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 modelaxreplem.3 . . 3 (𝜓 → ∅ ∈ 𝑀)
2 eleq1 2829 . . 3 (𝐴 = ∅ → (𝐴𝑀 ↔ ∅ ∈ 𝑀))
31, 2syl5ibrcom 247 . 2 (𝜓 → (𝐴 = ∅ → 𝐴𝑀))
4 vex 3484 . . . . . 6 𝑥 ∈ V
5 modelaxreplem1.5 . . . . . 6 𝐴𝑥
64, 5ssexi 5322 . . . . 5 𝐴 ∈ V
760sdom 9147 . . . 4 (∅ ≺ 𝐴𝐴 ≠ ∅)
8 ssdomg 9040 . . . . . 6 (𝑥 ∈ V → (𝐴𝑥𝐴𝑥))
94, 5, 8mp2 9 . . . . 5 𝐴𝑥
10 fodomr 9168 . . . . 5 ((∅ ≺ 𝐴𝐴𝑥) → ∃𝑔 𝑔:𝑥onto𝐴)
119, 10mpan2 691 . . . 4 (∅ ≺ 𝐴 → ∃𝑔 𝑔:𝑥onto𝐴)
127, 11sylbir 235 . . 3 (𝐴 ≠ ∅ → ∃𝑔 𝑔:𝑥onto𝐴)
13 df-fo 6567 . . . . 5 (𝑔:𝑥onto𝐴 ↔ (𝑔 Fn 𝑥 ∧ ran 𝑔 = 𝐴))
14 df-fn 6564 . . . . . . . 8 (𝑔 Fn 𝑥 ↔ (Fun 𝑔 ∧ dom 𝑔 = 𝑥))
15 modelaxreplem.4 . . . . . . . . . 10 (𝜓𝑥𝑀)
16 eleq1 2829 . . . . . . . . . 10 (dom 𝑔 = 𝑥 → (dom 𝑔𝑀𝑥𝑀))
1715, 16syl5ibrcom 247 . . . . . . . . 9 (𝜓 → (dom 𝑔 = 𝑥 → dom 𝑔𝑀))
1817anim2d 612 . . . . . . . 8 (𝜓 → ((Fun 𝑔 ∧ dom 𝑔 = 𝑥) → (Fun 𝑔 ∧ dom 𝑔𝑀)))
1914, 18biimtrid 242 . . . . . . 7 (𝜓 → (𝑔 Fn 𝑥 → (Fun 𝑔 ∧ dom 𝑔𝑀)))
20 modelaxreplem.1 . . . . . . . . 9 (𝜓𝑥𝑀)
215, 20sstrid 3995 . . . . . . . 8 (𝜓𝐴𝑀)
22 sseq1 4009 . . . . . . . 8 (ran 𝑔 = 𝐴 → (ran 𝑔𝑀𝐴𝑀))
2321, 22syl5ibrcom 247 . . . . . . 7 (𝜓 → (ran 𝑔 = 𝐴 → ran 𝑔𝑀))
24 df-3an 1089 . . . . . . . 8 ((Fun 𝑔 ∧ dom 𝑔𝑀 ∧ ran 𝑔𝑀) ↔ ((Fun 𝑔 ∧ dom 𝑔𝑀) ∧ ran 𝑔𝑀))
25 modelaxreplem.2 . . . . . . . . 9 (𝜓 → ∀𝑓((Fun 𝑓 ∧ dom 𝑓𝑀 ∧ ran 𝑓𝑀) → ran 𝑓𝑀))
26 funeq 6586 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (Fun 𝑓 ↔ Fun 𝑔))
27 dmeq 5914 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
2827eleq1d 2826 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (dom 𝑓𝑀 ↔ dom 𝑔𝑀))
29 rneq 5947 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
3029sseq1d 4015 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (ran 𝑓𝑀 ↔ ran 𝑔𝑀))
3126, 28, 303anbi123d 1438 . . . . . . . . . . 11 (𝑓 = 𝑔 → ((Fun 𝑓 ∧ dom 𝑓𝑀 ∧ ran 𝑓𝑀) ↔ (Fun 𝑔 ∧ dom 𝑔𝑀 ∧ ran 𝑔𝑀)))
3229eleq1d 2826 . . . . . . . . . . 11 (𝑓 = 𝑔 → (ran 𝑓𝑀 ↔ ran 𝑔𝑀))
3331, 32imbi12d 344 . . . . . . . . . 10 (𝑓 = 𝑔 → (((Fun 𝑓 ∧ dom 𝑓𝑀 ∧ ran 𝑓𝑀) → ran 𝑓𝑀) ↔ ((Fun 𝑔 ∧ dom 𝑔𝑀 ∧ ran 𝑔𝑀) → ran 𝑔𝑀)))
3433spvv 1996 . . . . . . . . 9 (∀𝑓((Fun 𝑓 ∧ dom 𝑓𝑀 ∧ ran 𝑓𝑀) → ran 𝑓𝑀) → ((Fun 𝑔 ∧ dom 𝑔𝑀 ∧ ran 𝑔𝑀) → ran 𝑔𝑀))
3525, 34syl 17 . . . . . . . 8 (𝜓 → ((Fun 𝑔 ∧ dom 𝑔𝑀 ∧ ran 𝑔𝑀) → ran 𝑔𝑀))
3624, 35biimtrrid 243 . . . . . . 7 (𝜓 → (((Fun 𝑔 ∧ dom 𝑔𝑀) ∧ ran 𝑔𝑀) → ran 𝑔𝑀))
3719, 23, 36syl2and 608 . . . . . 6 (𝜓 → ((𝑔 Fn 𝑥 ∧ ran 𝑔 = 𝐴) → ran 𝑔𝑀))
38 eleq1 2829 . . . . . . 7 (ran 𝑔 = 𝐴 → (ran 𝑔𝑀𝐴𝑀))
3938adantl 481 . . . . . 6 ((𝑔 Fn 𝑥 ∧ ran 𝑔 = 𝐴) → (ran 𝑔𝑀𝐴𝑀))
4037, 39mpbidi 241 . . . . 5 (𝜓 → ((𝑔 Fn 𝑥 ∧ ran 𝑔 = 𝐴) → 𝐴𝑀))
4113, 40biimtrid 242 . . . 4 (𝜓 → (𝑔:𝑥onto𝐴𝐴𝑀))
4241exlimdv 1933 . . 3 (𝜓 → (∃𝑔 𝑔:𝑥onto𝐴𝐴𝑀))
4312, 42syl5 34 . 2 (𝜓 → (𝐴 ≠ ∅ → 𝐴𝑀))
443, 43pm2.61dne 3028 1 (𝜓𝐴𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087  wal 1538   = wceq 1540  wex 1779  wcel 2108  wne 2940  Vcvv 3480  wss 3951  c0 4333   class class class wbr 5143  dom cdm 5685  ran crn 5686  Fun wfun 6555   Fn wfn 6556  ontowfo 6559  cdom 8983  csdm 8984
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 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
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 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-en 8986  df-dom 8987  df-sdom 8988
This theorem is referenced by:  modelaxreplem2  44996
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