Theorem rexrnmpt 7035

Description: A restricted quantifier over an image set. Usage of this theorem is discouraged because it depends on ax-13 2370. Use the weaker rexrnmptw 7033 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 7034	. . 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 3056	. 2 ⊢ (∃𝑦 ∈ ran 𝐹𝜓 ↔ ¬ ∀𝑦 ∈ ran 𝐹 ¬ 𝜓)
7		dfrex2 3056	. 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 3044  ∃wrex 3053   ↦ cmpt 5176  ran crn 5624

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 2370  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-fv 6494

This theorem is referenced by: (None)