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Mirrors > Home > MPE Home > Th. List > relmptopab | Structured version Visualization version GIF version |
Description: Any function to sets of ordered pairs produces a relation on function value unconditionally. (Contributed by Mario Carneiro, 7-Aug-2014.) (Proof shortened by Mario Carneiro, 24-Dec-2016.) |
Ref | Expression |
---|---|
relmptopab.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {〈𝑦, 𝑧〉 ∣ 𝜑}) |
Ref | Expression |
---|---|
relmptopab | ⊢ Rel (𝐹‘𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relmptopab.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {〈𝑦, 𝑧〉 ∣ 𝜑}) | |
2 | 1 | fvmptss 6772 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 {〈𝑦, 𝑧〉 ∣ 𝜑} ⊆ (V × V) → (𝐹‘𝐵) ⊆ (V × V)) |
3 | relopab 5689 | . . . . 5 ⊢ Rel {〈𝑦, 𝑧〉 ∣ 𝜑} | |
4 | df-rel 5555 | . . . . 5 ⊢ (Rel {〈𝑦, 𝑧〉 ∣ 𝜑} ↔ {〈𝑦, 𝑧〉 ∣ 𝜑} ⊆ (V × V)) | |
5 | 3, 4 | mpbi 231 | . . . 4 ⊢ {〈𝑦, 𝑧〉 ∣ 𝜑} ⊆ (V × V) |
6 | 5 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐴 → {〈𝑦, 𝑧〉 ∣ 𝜑} ⊆ (V × V)) |
7 | 2, 6 | mprg 3149 | . 2 ⊢ (𝐹‘𝐵) ⊆ (V × V) |
8 | df-rel 5555 | . 2 ⊢ (Rel (𝐹‘𝐵) ↔ (𝐹‘𝐵) ⊆ (V × V)) | |
9 | 7, 8 | mpbir 232 | 1 ⊢ Rel (𝐹‘𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 Vcvv 3492 ⊆ wss 3933 {copab 5119 ↦ cmpt 5137 × cxp 5546 Rel wrel 5553 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fv 6356 |
This theorem is referenced by: reldvdsr 19323 lmrel 21766 phtpcrel 23524 ulmrel 24893 ercgrg 26230 relwlk 27334 reltrls 27403 relpths 27428 releupth 27905 acycgr0v 32292 prclisacycgr 32295 |
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