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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 7000 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 {〈𝑦, 𝑧〉 ∣ 𝜑} ⊆ (V × V) → (𝐹‘𝐵) ⊆ (V × V)) |
3 | relopab 5814 | . . . . 5 ⊢ Rel {〈𝑦, 𝑧〉 ∣ 𝜑} | |
4 | df-rel 5673 | . . . . 5 ⊢ (Rel {〈𝑦, 𝑧〉 ∣ 𝜑} ↔ {〈𝑦, 𝑧〉 ∣ 𝜑} ⊆ (V × V)) | |
5 | 3, 4 | mpbi 229 | . . . 4 ⊢ {〈𝑦, 𝑧〉 ∣ 𝜑} ⊆ (V × V) |
6 | 5 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐴 → {〈𝑦, 𝑧〉 ∣ 𝜑} ⊆ (V × V)) |
7 | 2, 6 | mprg 3059 | . 2 ⊢ (𝐹‘𝐵) ⊆ (V × V) |
8 | df-rel 5673 | . 2 ⊢ (Rel (𝐹‘𝐵) ↔ (𝐹‘𝐵) ⊆ (V × V)) | |
9 | 7, 8 | mpbir 230 | 1 ⊢ Rel (𝐹‘𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 Vcvv 3466 ⊆ wss 3940 {copab 5200 ↦ cmpt 5221 × cxp 5664 Rel wrel 5671 ‘cfv 6533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fv 6541 |
This theorem is referenced by: reldvdsr 20251 lmrel 23055 phtpcrel 24840 ulmrel 26230 ercgrg 28203 relwlk 29318 reltrls 29386 relpths 29412 releupth 29887 acycgr0v 34594 prclisacycgr 34597 |
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