Theorem fvopabf4g 37684

Description: Function value of an operator abstraction whose domain is a set of functions with given domain and range. (Contributed by Jeff Madsen, 1-Dec-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)

Hypotheses

Ref	Expression
fvopabf4g.1	⊢ 𝐶 ∈ V
fvopabf4g.2	⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
fvopabf4g.3	⊢ 𝐹 = (𝑥 ∈ (𝑅 ↑m 𝐷) ↦ 𝐵)

Assertion

Ref	Expression
fvopabf4g	⊢ ((𝐷 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌 ∧ 𝐴:𝐷⟶𝑅) → (𝐹‘𝐴) = 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝑥,𝑅

Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)   𝑋(𝑥)   𝑌(𝑥)

Proof of Theorem fvopabf4g

Step	Hyp	Ref	Expression
1		elmapg 8899	. . . 4 ⊢ ((𝑅 ∈ 𝑌 ∧ 𝐷 ∈ 𝑋) → (𝐴 ∈ (𝑅 ↑m 𝐷) ↔ 𝐴:𝐷⟶𝑅))
2	1	ancoms 458	. . 3 ⊢ ((𝐷 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌) → (𝐴 ∈ (𝑅 ↑m 𝐷) ↔ 𝐴:𝐷⟶𝑅))
3	2	biimp3ar 1470	. 2 ⊢ ((𝐷 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌 ∧ 𝐴:𝐷⟶𝑅) → 𝐴 ∈ (𝑅 ↑m 𝐷))
4		fvopabf4g.2	. . 3 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
5		fvopabf4g.3	. . 3 ⊢ 𝐹 = (𝑥 ∈ (𝑅 ↑m 𝐷) ↦ 𝐵)
6		fvopabf4g.1	. . 3 ⊢ 𝐶 ∈ V
7	4, 5, 6	fvmpt 7031	. 2 ⊢ (𝐴 ∈ (𝑅 ↑m 𝐷) → (𝐹‘𝐴) = 𝐶)
8	3, 7	syl 17	1 ⊢ ((𝐷 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌 ∧ 𝐴:𝐷⟶𝑅) → (𝐹‘𝐴) = 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ↦ cmpt 5249  ⟶wf 6571  ‘cfv 6575  (class class class)co 7450   ↑_m cmap 8886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-fv 6583  df-ov 7453  df-oprab 7454  df-mpo 7455  df-map 8888

This theorem is referenced by: (None)