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Mirrors > Home > MPE Home > Th. List > relmpoopab | 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, 9-Feb-2015.) |
Ref | Expression |
---|---|
relmpoopab.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {〈𝑧, 𝑤〉 ∣ 𝜑}) |
Ref | Expression |
---|---|
relmpoopab | ⊢ Rel (𝐶𝐹𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relopab 5696 | . . . . 5 ⊢ Rel {〈𝑧, 𝑤〉 ∣ 𝜑} | |
2 | df-rel 5562 | . . . . 5 ⊢ (Rel {〈𝑧, 𝑤〉 ∣ 𝜑} ↔ {〈𝑧, 𝑤〉 ∣ 𝜑} ⊆ (V × V)) | |
3 | 1, 2 | mpbi 232 | . . . 4 ⊢ {〈𝑧, 𝑤〉 ∣ 𝜑} ⊆ (V × V) |
4 | 3 | rgen2w 3151 | . . 3 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 {〈𝑧, 𝑤〉 ∣ 𝜑} ⊆ (V × V) |
5 | relmpoopab.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {〈𝑧, 𝑤〉 ∣ 𝜑}) | |
6 | 5 | ovmptss 7788 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 {〈𝑧, 𝑤〉 ∣ 𝜑} ⊆ (V × V) → (𝐶𝐹𝐷) ⊆ (V × V)) |
7 | 4, 6 | ax-mp 5 | . 2 ⊢ (𝐶𝐹𝐷) ⊆ (V × V) |
8 | df-rel 5562 | . 2 ⊢ (Rel (𝐶𝐹𝐷) ↔ (𝐶𝐹𝐷) ⊆ (V × V)) | |
9 | 7, 8 | mpbir 233 | 1 ⊢ Rel (𝐶𝐹𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∀wral 3138 Vcvv 3494 ⊆ wss 3936 {copab 5128 × cxp 5553 Rel wrel 5560 (class class class)co 7156 ∈ cmpo 7158 |
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 ax-un 7461 |
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-iun 4921 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 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 |
This theorem is referenced by: brovmpoex 7889 relfunc 17132 releqg 18327 |
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