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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 6780 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 {〈𝑦, 𝑧〉 ∣ 𝜑} ⊆ (V × V) → (𝐹‘𝐵) ⊆ (V × V)) |
3 | relopab 5696 | . . . . 5 ⊢ Rel {〈𝑦, 𝑧〉 ∣ 𝜑} | |
4 | df-rel 5562 | . . . . 5 ⊢ (Rel {〈𝑦, 𝑧〉 ∣ 𝜑} ↔ {〈𝑦, 𝑧〉 ∣ 𝜑} ⊆ (V × V)) | |
5 | 3, 4 | mpbi 232 | . . . 4 ⊢ {〈𝑦, 𝑧〉 ∣ 𝜑} ⊆ (V × V) |
6 | 5 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐴 → {〈𝑦, 𝑧〉 ∣ 𝜑} ⊆ (V × V)) |
7 | 2, 6 | mprg 3152 | . 2 ⊢ (𝐹‘𝐵) ⊆ (V × V) |
8 | df-rel 5562 | . 2 ⊢ (Rel (𝐹‘𝐵) ↔ (𝐹‘𝐵) ⊆ (V × V)) | |
9 | 7, 8 | mpbir 233 | 1 ⊢ Rel (𝐹‘𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 Vcvv 3494 ⊆ wss 3936 {copab 5128 ↦ cmpt 5146 × cxp 5553 Rel wrel 5560 ‘cfv 6355 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fv 6363 |
This theorem is referenced by: reldvdsr 19394 lmrel 21838 phtpcrel 23597 ulmrel 24966 ercgrg 26303 relwlk 27407 reltrls 27476 relpths 27501 releupth 27978 acycgr0v 32395 prclisacycgr 32398 |
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