Theorem fvopabf4g 38406

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 8845	. . . 4 ⊢ ((𝑅 ∈ 𝑌 ∧ 𝐷 ∈ 𝑋) → (𝐴 ∈ (𝑅 ↑m 𝐷) ↔ 𝐴:𝐷⟶𝑅))
2	1	ancoms 464	. . 3 ⊢ ((𝐷 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌) → (𝐴 ∈ (𝑅 ↑m 𝐷) ↔ 𝐴:𝐷⟶𝑅))
3	2	biimp3ar 1499	. 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 6996	. 2 ⊢ (𝐴 ∈ (𝑅 ↑m 𝐷) → (𝐹‘𝐴) = 𝐶)
8	3, 7	syl 18	1 ⊢ ((𝐷 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌 ∧ 𝐴:𝐷⟶𝑅) → (𝐹‘𝐴) = 𝐶)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  Vcvv 3458   ↦ cmpt 5197  ⟶wf 6539  ‘cfv 6543  (class class class)co 7423   ↑_m cmap 8833

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-map 8835

This theorem is used by: (None)