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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 2737 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 = 𝐶 ↔ 𝑦 = 𝐷)) |
5 | rnmptpr.e | . . . . . 6 ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸) | |
6 | 5 | eqeq2d 2737 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑦 = 𝐶 ↔ 𝑦 = 𝐸)) |
7 | 4, 6 | rexprg 4695 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷 ∨ 𝑦 = 𝐸))) |
8 | 1, 2, 7 | syl2anc 583 | . . 3 ⊢ (𝜑 → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷 ∨ 𝑦 = 𝐸))) |
9 | rnmptpr.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐶) | |
10 | 9 | elrnmpt 5948 | . . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶)) |
11 | 10 | elv 3474 | . . 3 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶) |
12 | vex 3472 | . . . 4 ⊢ 𝑦 ∈ V | |
13 | 12 | elpr 4646 | . . 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 2724 | 1 ⊢ (𝜑 → ran 𝐹 = {𝐷, 𝐸}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∨ wo 844 = wceq 1533 ∈ wcel 2098 ∃wrex 3064 Vcvv 3468 {cpr 4625 ↦ cmpt 5224 ran crn 5670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-mpt 5225 df-cnv 5677 df-dm 5679 df-rn 5680 |
This theorem is referenced by: sge0pr 45664 |
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