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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rnmptpr | Structured version Visualization version GIF version |
Description: Range of a function defined on an unordered pair. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
rnmptpr.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
rnmptpr.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
rnmptpr.f | ⊢ 𝐹 = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐶) |
rnmptpr.d | ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷) |
rnmptpr.e | ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸) |
Ref | Expression |
---|---|
rnmptpr | ⊢ (𝜑 → ran 𝐹 = {𝐷, 𝐸}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnmptpr.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | rnmptpr.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
3 | rnmptpr.d | . . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷) | |
4 | 3 | eqeq2d 2746 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 = 𝐶 ↔ 𝑦 = 𝐷)) |
5 | rnmptpr.e | . . . . . 6 ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸) | |
6 | 5 | eqeq2d 2746 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑦 = 𝐶 ↔ 𝑦 = 𝐸)) |
7 | 4, 6 | rexprg 4702 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷 ∨ 𝑦 = 𝐸))) |
8 | 1, 2, 7 | syl2anc 584 | . . 3 ⊢ (𝜑 → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷 ∨ 𝑦 = 𝐸))) |
9 | rnmptpr.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐶) | |
10 | 9 | elrnmpt 5972 | . . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶)) |
11 | 10 | elv 3483 | . . 3 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶) |
12 | vex 3482 | . . . 4 ⊢ 𝑦 ∈ V | |
13 | 12 | elpr 4655 | . . 3 ⊢ (𝑦 ∈ {𝐷, 𝐸} ↔ (𝑦 = 𝐷 ∨ 𝑦 = 𝐸)) |
14 | 8, 11, 13 | 3bitr4g 314 | . 2 ⊢ (𝜑 → (𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ {𝐷, 𝐸})) |
15 | 14 | eqrdv 2733 | 1 ⊢ (𝜑 → ran 𝐹 = {𝐷, 𝐸}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 Vcvv 3478 {cpr 4633 ↦ cmpt 5231 ran crn 5690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-mpt 5232 df-cnv 5697 df-dm 5699 df-rn 5700 |
This theorem is referenced by: sge0pr 46350 |
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