Theorem pw2f1o2val2 38387

Description: Membership in a mapped set under the pw2f1o2 38385 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)

Hypothesis

Ref	Expression
pw2f1o2.f	⊢ 𝐹 = (𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜}))

Assertion

Ref	Expression
pw2f1o2val2	⊢ ((𝑋 ∈ (2𝑜 ↑𝑚 𝐴) ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑋‘𝑌) = 1𝑜))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑋   𝑥,𝑌

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem pw2f1o2val2

Step	Hyp	Ref	Expression
1		pw2f1o2.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜}))
2	1	pw2f1o2val 38386	. . . 4 ⊢ (𝑋 ∈ (2𝑜 ↑𝑚 𝐴) → (𝐹‘𝑋) = (◡𝑋 “ {1𝑜}))
3	2	eleq2d 2865	. . 3 ⊢ (𝑋 ∈ (2𝑜 ↑𝑚 𝐴) → (𝑌 ∈ (𝐹‘𝑋) ↔ 𝑌 ∈ (◡𝑋 “ {1𝑜})))
4	3	adantr 473	. 2 ⊢ ((𝑋 ∈ (2𝑜 ↑𝑚 𝐴) ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (𝐹‘𝑋) ↔ 𝑌 ∈ (◡𝑋 “ {1𝑜})))
5		elmapi 8118	. . . 4 ⊢ (𝑋 ∈ (2𝑜 ↑𝑚 𝐴) → 𝑋:𝐴⟶2𝑜)
6		ffn 6257	. . . 4 ⊢ (𝑋:𝐴⟶2𝑜 → 𝑋 Fn 𝐴)
7		fniniseg 6565	. . . 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 536	. 2 ⊢ ((𝑋 ∈ (2𝑜 ↑𝑚 𝐴) ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (◡𝑋 “ {1𝑜}) ↔ (𝑋‘𝑌) = 1𝑜))
10	4, 9	bitrd 271	1 ⊢ ((𝑋 ∈ (2𝑜 ↑𝑚 𝐴) ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑋‘𝑌) = 1𝑜))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   = wceq 1653   ∈ wcel 2157  {csn 4369   ↦ cmpt 4923  ^◡ccnv 5312   “ cima 5316   Fn wfn 6097  ⟶wf 6098  ‘cfv 6102  (class class class)co 6879  1_𝑜c1o 7793  2_𝑜c2o 7794   ↑_𝑚 cmap 8096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-fv 6110  df-ov 6882  df-oprab 6883  df-mpt2 6884  df-1st 7402  df-2nd 7403  df-map 8098

This theorem is referenced by:  wepwsolem  38392