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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 5689 | . . . . 5 ⊢ Rel {〈𝑧, 𝑤〉 ∣ 𝜑} | |
2 | df-rel 5555 | . . . . 5 ⊢ (Rel {〈𝑧, 𝑤〉 ∣ 𝜑} ↔ {〈𝑧, 𝑤〉 ∣ 𝜑} ⊆ (V × V)) | |
3 | 1, 2 | mpbi 232 | . . . 4 ⊢ {〈𝑧, 𝑤〉 ∣ 𝜑} ⊆ (V × V) |
4 | 3 | rgen2w 3150 | . . 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 7781 | . . 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 5555 | . 2 ⊢ (Rel (𝐶𝐹𝐷) ↔ (𝐶𝐹𝐷) ⊆ (V × V)) | |
9 | 7, 8 | mpbir 233 | 1 ⊢ Rel (𝐶𝐹𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∀wral 3137 Vcvv 3491 ⊆ wss 3929 {copab 5121 × cxp 5546 Rel wrel 5553 (class class class)co 7149 ∈ cmpo 7151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 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 df-ov 7152 df-oprab 7153 df-mpo 7154 df-1st 7682 df-2nd 7683 |
This theorem is referenced by: brovmpoex 7882 relfunc 17125 releqg 18320 |
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