Theorem rexrnmptw 7114

Description: A restricted quantifier over an image set. Version of rexrnmpt 7116 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by Mario Carneiro, 20-Aug-2015.) Avoid ax-13 2374. (Revised by GG, 26-Jan-2024.)

Hypotheses

Ref	Expression
rexrnmptw.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
rexrnmptw.2	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
rexrnmptw	⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))

Distinct variable groups:   𝑥,𝑦   𝑥,𝐴   𝑦,𝐵   𝜒,𝑦   𝑦,𝐹   𝜓,𝑥

Allowed substitution hints:   𝜓(𝑦)   𝜒(𝑥)   𝐴(𝑦)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem rexrnmptw

Step	Hyp	Ref	Expression
1		rexrnmptw.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
2		rexrnmptw.2	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
3	2	notbid 318	. . . 4 ⊢ (𝑦 = 𝐵 → (¬ 𝜓 ↔ ¬ 𝜒))
4	1, 3	ralrnmptw 7113	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑦 ∈ ran 𝐹 ¬ 𝜓 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝜒))
5	4	notbid 318	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (¬ ∀𝑦 ∈ ran 𝐹 ¬ 𝜓 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜒))
6		dfrex2 3070	. 2 ⊢ (∃𝑦 ∈ ran 𝐹𝜓 ↔ ¬ ∀𝑦 ∈ ran 𝐹 ¬ 𝜓)
7		dfrex2 3070	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜒 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜒)
8	5, 6, 7	3bitr4g 314	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1536   ∈ wcel 2105  ∀wral 3058  ∃wrex 3067   ↦ cmpt 5230  ran crn 5689

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-sep 5301  ax-nul 5311  ax-pr 5437

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-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-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  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-fv 6570

This theorem is referenced by:  onoviun  8381  onnseq  8382  ghmcyg  19928  pgpfac1lem2  20109  pgpfac1lem3  20111  pgpfac1lem4  20112  pptbas  23030  lly1stc  23519  txbas  23590  eltsms  24156  tsmsf1o  24168  psmetutop  24595  xrge0tsms  24869  fmcfil  25319  ellimc2  25926  limcflf  25930  xrge0tsmsd  33047  rspectopn  33827  poimirlem23  37629  poimirlem24  37630  poimirlem30  37636  cntotbnd  37782  mnurndlem1  44276