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Mirrors > Home > MPE Home > Th. List > fnopfv | Structured version Visualization version GIF version |
Description: Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 30-Sep-2004.) |
Ref | Expression |
---|---|
fnopfv | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 〈𝐵, (𝐹‘𝐵)〉 ∈ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funfvop 6578 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → 〈𝐵, (𝐹‘𝐵)〉 ∈ 𝐹) | |
2 | 1 | funfni 6224 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 〈𝐵, (𝐹‘𝐵)〉 ∈ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2166 〈cop 4403 Fn wfn 6118 ‘cfv 6123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-iota 6086 df-fun 6125 df-fn 6126 df-fv 6131 |
This theorem is referenced by: foeqcnvco 6810 |
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