![]() |
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 | vex 3418 | . . . . . 6 ⊢ 𝑦 ∈ V | |
2 | rnmptpr.f | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐶) | |
3 | 2 | elrnmpt 5606 | . . . . . 6 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶)) |
4 | 1, 3 | ax-mp 5 | . . . . 5 ⊢ (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶) |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶)) |
6 | rnmptpr.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
7 | rnmptpr.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
8 | rnmptpr.d | . . . . . . 7 ⊢ (𝑥 = 𝐴 → 𝐶 = 𝐷) | |
9 | 8 | eqeq2d 2836 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑦 = 𝐶 ↔ 𝑦 = 𝐷)) |
10 | rnmptpr.e | . . . . . . 7 ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸) | |
11 | 10 | eqeq2d 2836 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑦 = 𝐶 ↔ 𝑦 = 𝐸)) |
12 | 9, 11 | rexprg 4455 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷 ∨ 𝑦 = 𝐸))) |
13 | 6, 7, 12 | syl2anc 581 | . . . 4 ⊢ (𝜑 → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷 ∨ 𝑦 = 𝐸))) |
14 | 1 | elpr 4421 | . . . . . 6 ⊢ (𝑦 ∈ {𝐷, 𝐸} ↔ (𝑦 = 𝐷 ∨ 𝑦 = 𝐸)) |
15 | 14 | bicomi 216 | . . . . 5 ⊢ ((𝑦 = 𝐷 ∨ 𝑦 = 𝐸) ↔ 𝑦 ∈ {𝐷, 𝐸}) |
16 | 15 | a1i 11 | . . . 4 ⊢ (𝜑 → ((𝑦 = 𝐷 ∨ 𝑦 = 𝐸) ↔ 𝑦 ∈ {𝐷, 𝐸})) |
17 | 5, 13, 16 | 3bitrd 297 | . . 3 ⊢ (𝜑 → (𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ {𝐷, 𝐸})) |
18 | 17 | alrimiv 2028 | . 2 ⊢ (𝜑 → ∀𝑦(𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ {𝐷, 𝐸})) |
19 | dfcleq 2820 | . 2 ⊢ (ran 𝐹 = {𝐷, 𝐸} ↔ ∀𝑦(𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ {𝐷, 𝐸})) | |
20 | 18, 19 | sylibr 226 | 1 ⊢ (𝜑 → ran 𝐹 = {𝐷, 𝐸}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∨ wo 880 ∀wal 1656 = wceq 1658 ∈ wcel 2166 ∃wrex 3119 Vcvv 3415 {cpr 4400 ↦ cmpt 4953 ran crn 5344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4875 df-opab 4937 df-mpt 4954 df-cnv 5351 df-dm 5353 df-rn 5354 |
This theorem is referenced by: sge0pr 41403 |
Copyright terms: Public domain | W3C validator |