Theorem rexrnmpt 7078

Description: A restricted quantifier over an image set. Usage of this theorem is discouraged because it depends on ax-13 2403. Use the weaker rexrnmptw 7076 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 320	. . . 4 ⊢ (𝑦 = 𝐵 → (¬ 𝜓 ↔ ¬ 𝜒))
4	1, 3	ralrnmpt 7077	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑦 ∈ ran 𝐹 ¬ 𝜓 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝜒))
5	4	notbid 320	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (¬ ∀𝑦 ∈ ran 𝐹 ¬ 𝜓 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜒))
6		dfrex2 3089	. 2 ⊢ (∃𝑦 ∈ ran 𝐹𝜓 ↔ ¬ ∀𝑦 ∈ ran 𝐹 ¬ 𝜓)
7		dfrex2 3089	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜒 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜒)
8	5, 6, 7	3bitr4g 316	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   = wceq 1560   ∈ wcel 2142  ∀wral 3076  ∃wrex 3086   ↦ cmpt 5181  ran crn 5648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-13 2403  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-fv 6529

This theorem is referenced by: (None)