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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 | relopabv 5817 | . . . . 5 ⊢ Rel {⟨𝑧, 𝑤⟩ ∣ 𝜑} | |
2 | df-rel 5679 | . . . . 5 ⊢ (Rel {⟨𝑧, 𝑤⟩ ∣ 𝜑} ↔ {⟨𝑧, 𝑤⟩ ∣ 𝜑} ⊆ (V × V)) | |
3 | 1, 2 | mpbi 229 | . . . 4 ⊢ {⟨𝑧, 𝑤⟩ ∣ 𝜑} ⊆ (V × V) |
4 | 3 | rgen2w 3056 | . . 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 8094 | . . 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 5679 | . 2 ⊢ (Rel (𝐶𝐹𝐷) ↔ (𝐶𝐹𝐷) ⊆ (V × V)) | |
9 | 7, 8 | mpbir 230 | 1 ⊢ Rel (𝐶𝐹𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∀wral 3051 Vcvv 3463 ⊆ wss 3940 {copab 5205 × cxp 5670 Rel wrel 5677 (class class class)co 7415 ∈ cmpo 7417 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7737 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fv 6550 df-ov 7418 df-oprab 7419 df-mpo 7420 df-1st 7989 df-2nd 7990 |
This theorem is referenced by: brovmpoex 8225 relfunc 17845 releqg 19132 |
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