Theorem rexrnmpt 7112

Description: A restricted quantifier over an image set. Usage of this theorem is discouraged because it depends on ax-13 2366. Use the weaker rexrnmptw 7110 when possible. (Contributed by Mario Carneiro, 20-Aug-2015.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

Distinct variable groups:   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 rexrnmpt

Step	Hyp	Ref	Expression
1		ralrnmpt.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
2		ralrnmpt.2	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
3	2	notbid 317	. . . 4 ⊢ (𝑦 = 𝐵 → (¬ 𝜓 ↔ ¬ 𝜒))
4	1, 3	ralrnmpt 7111	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑦 ∈ ran 𝐹 ¬ 𝜓 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝜒))
5	4	notbid 317	. 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 313	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  ∀wral 3058  ∃wrex 3067   ↦ cmpt 5235  ran crn 5683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-13 2366  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-fv 6561

This theorem is referenced by: (None)