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Mirrors > Home > MPE Home > Th. List > xpsfrnel2 | Structured version Visualization version GIF version |
Description: Elementhood in the target space of the function 𝐹 appearing in xpsval 16842. (Contributed by Mario Carneiro, 15-Aug-2015.) |
Ref | Expression |
---|---|
xpsfrnel2 | ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpsfrnel 16834 | . 2 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ∧ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘∅) ∈ 𝐴 ∧ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘1o) ∈ 𝐵)) | |
2 | fnpr2ob 16830 | . . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ↔ {〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o) | |
3 | 2 | biimpri 230 | . . . 4 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o → (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
4 | 3 | 3ad2ant1 1129 | . . 3 ⊢ (({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ∧ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘∅) ∈ 𝐴 ∧ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘1o) ∈ 𝐵) → (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
5 | elex 3512 | . . . 4 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ V) | |
6 | elex 3512 | . . . 4 ⊢ (𝑌 ∈ 𝐵 → 𝑌 ∈ V) | |
7 | 5, 6 | anim12i 614 | . . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
8 | 3anass 1091 | . . . 4 ⊢ (({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ∧ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘∅) ∈ 𝐴 ∧ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘1o) ∈ 𝐵) ↔ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ∧ (({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘∅) ∈ 𝐴 ∧ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘1o) ∈ 𝐵))) | |
9 | fnpr2o 16829 | . . . . . 6 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → {〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o) | |
10 | 9 | biantrurd 535 | . . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → ((({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘∅) ∈ 𝐴 ∧ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘1o) ∈ 𝐵) ↔ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ∧ (({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘∅) ∈ 𝐴 ∧ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘1o) ∈ 𝐵)))) |
11 | fvpr0o 16831 | . . . . . . 7 ⊢ (𝑋 ∈ V → ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘∅) = 𝑋) | |
12 | 11 | eleq1d 2897 | . . . . . 6 ⊢ (𝑋 ∈ V → (({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘∅) ∈ 𝐴 ↔ 𝑋 ∈ 𝐴)) |
13 | fvpr1o 16832 | . . . . . . 7 ⊢ (𝑌 ∈ V → ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘1o) = 𝑌) | |
14 | 13 | eleq1d 2897 | . . . . . 6 ⊢ (𝑌 ∈ V → (({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘1o) ∈ 𝐵 ↔ 𝑌 ∈ 𝐵)) |
15 | 12, 14 | bi2anan9 637 | . . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → ((({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘∅) ∈ 𝐴 ∧ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘1o) ∈ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵))) |
16 | 10, 15 | bitr3d 283 | . . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ∧ (({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘∅) ∈ 𝐴 ∧ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘1o) ∈ 𝐵)) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵))) |
17 | 8, 16 | syl5bb 285 | . . 3 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ∧ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘∅) ∈ 𝐴 ∧ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘1o) ∈ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵))) |
18 | 4, 7, 17 | pm5.21nii 382 | . 2 ⊢ (({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ∧ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘∅) ∈ 𝐴 ∧ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘1o) ∈ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) |
19 | 1, 18 | bitri 277 | 1 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∅c0 4290 ifcif 4466 {cpr 4568 〈cop 4572 Fn wfn 6349 ‘cfv 6354 1oc1o 8094 2oc2o 8095 Xcixp 8460 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 |
This theorem is referenced by: xpscf 16837 xpsff1o 16839 |
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