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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pw2f1o2val2 | Structured version Visualization version GIF version |
Description: Membership in a mapped set under the pw2f1o2 41405 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
Ref | Expression |
---|---|
pw2f1o2.f | ⊢ 𝐹 = (𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o})) |
Ref | Expression |
---|---|
pw2f1o2val2 | ⊢ ((𝑋 ∈ (2o ↑m 𝐴) ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑋‘𝑌) = 1o)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pw2f1o2.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o})) | |
2 | 1 | pw2f1o2val 41406 | . . . 4 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → (𝐹‘𝑋) = (◡𝑋 “ {1o})) |
3 | 2 | eleq2d 2820 | . . 3 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → (𝑌 ∈ (𝐹‘𝑋) ↔ 𝑌 ∈ (◡𝑋 “ {1o}))) |
4 | 3 | adantr 482 | . 2 ⊢ ((𝑋 ∈ (2o ↑m 𝐴) ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (𝐹‘𝑋) ↔ 𝑌 ∈ (◡𝑋 “ {1o}))) |
5 | elmapi 8790 | . . . 4 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → 𝑋:𝐴⟶2o) | |
6 | ffn 6669 | . . . 4 ⊢ (𝑋:𝐴⟶2o → 𝑋 Fn 𝐴) | |
7 | fniniseg 7011 | . . . 4 ⊢ (𝑋 Fn 𝐴 → (𝑌 ∈ (◡𝑋 “ {1o}) ↔ (𝑌 ∈ 𝐴 ∧ (𝑋‘𝑌) = 1o))) | |
8 | 5, 6, 7 | 3syl 18 | . . 3 ⊢ (𝑋 ∈ (2o ↑m 𝐴) → (𝑌 ∈ (◡𝑋 “ {1o}) ↔ (𝑌 ∈ 𝐴 ∧ (𝑋‘𝑌) = 1o))) |
9 | 8 | baibd 541 | . 2 ⊢ ((𝑋 ∈ (2o ↑m 𝐴) ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (◡𝑋 “ {1o}) ↔ (𝑋‘𝑌) = 1o)) |
10 | 4, 9 | bitrd 279 | 1 ⊢ ((𝑋 ∈ (2o ↑m 𝐴) ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑋‘𝑌) = 1o)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {csn 4587 ↦ cmpt 5189 ◡ccnv 5633 “ cima 5637 Fn wfn 6492 ⟶wf 6493 ‘cfv 6497 (class class class)co 7358 1oc1o 8406 2oc2o 8407 ↑m cmap 8768 |
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 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-map 8770 |
This theorem is referenced by: wepwsolem 41412 |
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