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Mirrors > Home > MPE Home > Th. List > relmpt2opab | 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 |
---|---|
relmpt2opab.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {〈𝑧, 𝑤〉 ∣ 𝜑}) |
Ref | Expression |
---|---|
relmpt2opab | ⊢ Rel (𝐶𝐹𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relopab 5169 | . . . . 5 ⊢ Rel {〈𝑧, 𝑤〉 ∣ 𝜑} | |
2 | df-rel 5045 | . . . . 5 ⊢ (Rel {〈𝑧, 𝑤〉 ∣ 𝜑} ↔ {〈𝑧, 𝑤〉 ∣ 𝜑} ⊆ (V × V)) | |
3 | 1, 2 | mpbi 219 | . . . 4 ⊢ {〈𝑧, 𝑤〉 ∣ 𝜑} ⊆ (V × V) |
4 | 3 | rgen2w 2909 | . . 3 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 {〈𝑧, 𝑤〉 ∣ 𝜑} ⊆ (V × V) |
5 | relmpt2opab.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {〈𝑧, 𝑤〉 ∣ 𝜑}) | |
6 | 5 | ovmptss 7145 | . . 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 5045 | . 2 ⊢ (Rel (𝐶𝐹𝐷) ↔ (𝐶𝐹𝐷) ⊆ (V × V)) | |
9 | 7, 8 | mpbir 220 | 1 ⊢ Rel (𝐶𝐹𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∀wral 2896 Vcvv 3173 ⊆ wss 3540 {copab 4642 × cxp 5036 Rel wrel 5043 (class class class)co 6549 ↦ cmpt2 6551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 |
This theorem is referenced by: brovmpt2ex 7236 relfunc 16345 releqg 17464 relwlk 26046 releupa 26491 |
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