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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 317 | . . . 4 ⊢ (𝑧 = 𝐶 → (¬ 𝜑 ↔ ¬ 𝜓)) |
4 | 1, 3 | ralrnmpo 7549 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∀𝑧 ∈ ran 𝐹 ¬ 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜓)) |
5 | 4 | notbid 317 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (¬ ∀𝑧 ∈ ran 𝐹 ¬ 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜓)) |
6 | dfrex2 3073 | . 2 ⊢ (∃𝑧 ∈ ran 𝐹𝜑 ↔ ¬ ∀𝑧 ∈ ran 𝐹 ¬ 𝜑) | |
7 | dfrex2 3073 | . . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜓) | |
8 | 7 | rexbii 3094 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜓) |
9 | rexnal 3100 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜓 ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜓) | |
10 | 8, 9 | bitri 274 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 ↔ ¬ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜓) |
11 | 5, 6, 10 | 3bitr4g 313 | 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 205 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 ran crn 5677 ∈ cmpo 7413 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-cnv 5684 df-dm 5686 df-rn 5687 df-oprab 7415 df-mpo 7416 |
This theorem is referenced by: lsmass 19539 eltx 23079 txrest 23142 txlm 23159 lsmssass 32557 ptrest 36573 |
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