Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 2834 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 = 𝐶 ↔ 𝑦 = 𝐷)) |
5 | rnmptpr.e | . . . . . 6 ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸) | |
6 | 5 | eqeq2d 2834 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑦 = 𝐶 ↔ 𝑦 = 𝐸)) |
7 | 4, 6 | rexprg 4635 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷 ∨ 𝑦 = 𝐸))) |
8 | 1, 2, 7 | syl2anc 586 | . . 3 ⊢ (𝜑 → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷 ∨ 𝑦 = 𝐸))) |
9 | rnmptpr.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐶) | |
10 | 9 | elrnmpt 5830 | . . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶)) |
11 | 10 | elv 3501 | . . 3 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶) |
12 | vex 3499 | . . . 4 ⊢ 𝑦 ∈ V | |
13 | 12 | elpr 4592 | . . 3 ⊢ (𝑦 ∈ {𝐷, 𝐸} ↔ (𝑦 = 𝐷 ∨ 𝑦 = 𝐸)) |
14 | 8, 11, 13 | 3bitr4g 316 | . 2 ⊢ (𝜑 → (𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ {𝐷, 𝐸})) |
15 | 14 | eqrdv 2821 | 1 ⊢ (𝜑 → ran 𝐹 = {𝐷, 𝐸}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 Vcvv 3496 {cpr 4571 ↦ cmpt 5148 ran crn 5558 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-mpt 5149 df-cnv 5565 df-dm 5567 df-rn 5568 |
This theorem is referenced by: sge0pr 42683 |
Copyright terms: Public domain | W3C validator |