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Mirrors > Home > MPE Home > Th. List > rexrnmpo | Structured version Visualization version GIF version |
Description: A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.) |
Ref | Expression |
---|---|
rngop.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
ralrnmpo.2 | ⊢ (𝑧 = 𝐶 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
rexrnmpo | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngop.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
2 | ralrnmpo.2 | . . . . 5 ⊢ (𝑧 = 𝐶 → (𝜑 ↔ 𝜓)) | |
3 | 2 | notbid 320 | . . . 4 ⊢ (𝑧 = 𝐶 → (¬ 𝜑 ↔ ¬ 𝜓)) |
4 | 1, 3 | ralrnmpo 7289 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∀𝑧 ∈ ran 𝐹 ¬ 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜓)) |
5 | 4 | notbid 320 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (¬ ∀𝑧 ∈ ran 𝐹 ¬ 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜓)) |
6 | dfrex2 3239 | . 2 ⊢ (∃𝑧 ∈ ran 𝐹𝜑 ↔ ¬ ∀𝑧 ∈ ran 𝐹 ¬ 𝜑) | |
7 | dfrex2 3239 | . . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜓) | |
8 | 7 | rexbii 3247 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜓) |
9 | rexnal 3238 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜓 ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜓) | |
10 | 8, 9 | bitri 277 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜓) |
11 | 5, 6, 10 | 3bitr4g 316 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 ran crn 5556 ∈ cmpo 7158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-cnv 5563 df-dm 5565 df-rn 5566 df-oprab 7160 df-mpo 7161 |
This theorem is referenced by: lsmass 18795 eltx 22176 txrest 22239 txlm 22256 ptrest 34906 |
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