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 | vex 3176 | . . . . . 6 ⊢ 𝑦 ∈ V | |
2 | rnmptpr.f | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐶) | |
3 | 2 | elrnmpt 5293 | . . . . . 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 2620 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑦 = 𝐶 ↔ 𝑦 = 𝐷)) |
10 | rnmptpr.e | . . . . . . 7 ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐸) | |
11 | 10 | eqeq2d 2620 | . . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑦 = 𝐶 ↔ 𝑦 = 𝐸)) |
12 | 9, 11 | rexprg 4182 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷 ∨ 𝑦 = 𝐸))) |
13 | 6, 7, 12 | syl2anc 691 | . . . 4 ⊢ (𝜑 → (∃𝑥 ∈ {𝐴, 𝐵}𝑦 = 𝐶 ↔ (𝑦 = 𝐷 ∨ 𝑦 = 𝐸))) |
14 | 1 | elpr 4146 | . . . . . 6 ⊢ (𝑦 ∈ {𝐷, 𝐸} ↔ (𝑦 = 𝐷 ∨ 𝑦 = 𝐸)) |
15 | 14 | bicomi 213 | . . . . 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 293 | . . 3 ⊢ (𝜑 → (𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ {𝐷, 𝐸})) |
18 | 17 | alrimiv 1842 | . 2 ⊢ (𝜑 → ∀𝑦(𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ {𝐷, 𝐸})) |
19 | dfcleq 2604 | . 2 ⊢ (ran 𝐹 = {𝐷, 𝐸} ↔ ∀𝑦(𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ {𝐷, 𝐸})) | |
20 | 18, 19 | sylibr 223 | 1 ⊢ (𝜑 → ran 𝐹 = {𝐷, 𝐸}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∨ wo 382 ∀wal 1473 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 Vcvv 3173 {cpr 4127 ↦ cmpt 4643 ran crn 5039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-mpt 4645 df-cnv 5046 df-dm 5048 df-rn 5049 |
This theorem is referenced by: sge0pr 39287 |
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