Theorem rexrnmpt 7092

Description: A restricted quantifier over an image set. Usage of this theorem is discouraged because it depends on ax-13 2377. Use the weaker rexrnmptw 7090 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 318	. . . 4 ⊢ (𝑦 = 𝐵 → (¬ 𝜓 ↔ ¬ 𝜒))
4	1, 3	ralrnmpt 7091	. . 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 3064	. 2 ⊢ (∃𝑦 ∈ ran 𝐹𝜓 ↔ ¬ ∀𝑦 ∈ ran 𝐹 ¬ 𝜓)
7		dfrex2 3064	. 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 1540   ∈ wcel 2109  ∀wral 3052  ∃wrex 3061   ↦ cmpt 5206  ran crn 5660

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-13 2377  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-fv 6544

This theorem is referenced by: (None)