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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 5782 | . . . . 5 ⊢ Rel {⟨𝑧, 𝑤⟩ ∣ 𝜑} | |
2 | df-rel 5645 | . . . . 5 ⊢ (Rel {⟨𝑧, 𝑤⟩ ∣ 𝜑} ↔ {⟨𝑧, 𝑤⟩ ∣ 𝜑} ⊆ (V × V)) | |
3 | 1, 2 | mpbi 229 | . . . 4 ⊢ {⟨𝑧, 𝑤⟩ ∣ 𝜑} ⊆ (V × V) |
4 | 3 | rgen2w 3070 | . . 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 8030 | . . 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 5645 | . 2 ⊢ (Rel (𝐶𝐹𝐷) ↔ (𝐶𝐹𝐷) ⊆ (V × V)) | |
9 | 7, 8 | mpbir 230 | 1 ⊢ Rel (𝐶𝐹𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∀wral 3065 Vcvv 3448 ⊆ wss 3915 {copab 5172 × cxp 5636 Rel wrel 5643 (class class class)co 7362 ∈ cmpo 7364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-1st 7926 df-2nd 7927 |
This theorem is referenced by: brovmpoex 8159 relfunc 17755 releqg 18984 |
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