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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 2748 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 = 𝐶 ↔ 𝑦 = 𝐷)) |
| 5 | rnmptpr.e | . . . . . 6 ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸) | |
| 6 | 5 | eqeq2d 2748 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑦 = 𝐶 ↔ 𝑦 = 𝐸)) |
| 7 | 4, 6 | rexprg 4697 | . . . 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 5969 | . . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶)) |
| 11 | 10 | elv 3485 | . . 3 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶) |
| 12 | vex 3484 | . . . 4 ⊢ 𝑦 ∈ V | |
| 13 | 12 | elpr 4650 | . . 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 2735 | 1 ⊢ (𝜑 → ran 𝐹 = {𝐷, 𝐸}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 {cpr 4628 ↦ cmpt 5225 ran crn 5686 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-mpt 5226 df-cnv 5693 df-dm 5695 df-rn 5696 |
| This theorem is referenced by: sge0pr 46409 |
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