Theorem fliftval 7243

Description: The value of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)

Hypotheses

Ref	Expression
flift.1	⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)
flift.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)
flift.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)
fliftval.4	⊢ (𝑥 = 𝑌 → 𝐴 = 𝐶)
fliftval.5	⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷)
fliftval.6	⊢ (𝜑 → Fun 𝐹)

Assertion

Ref	Expression
fliftval	⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → (𝐹‘𝐶) = 𝐷)

Distinct variable groups:   𝑥,𝐶   𝑥,𝑅   𝑥,𝑌   𝑥,𝐷   𝜑,𝑥   𝑥,𝑋   𝑥,𝑆

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fliftval

Step	Hyp	Ref	Expression
1		fliftval.6	. . 3 ⊢ (𝜑 → Fun 𝐹)
2	1	adantr 481	. 2 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → Fun 𝐹)
3		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → 𝑌 ∈ 𝑋)
4		eqidd 2737	. . . . 5 ⊢ (𝜑 → 𝐷 = 𝐷)
5		eqidd 2737	. . . . 5 ⊢ (𝑌 ∈ 𝑋 → 𝐶 = 𝐶)
6	4, 5	anim12ci 614	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → (𝐶 = 𝐶 ∧ 𝐷 = 𝐷))
7		fliftval.4	. . . . . . 7 ⊢ (𝑥 = 𝑌 → 𝐴 = 𝐶)
8	7	eqeq2d 2747	. . . . . 6 ⊢ (𝑥 = 𝑌 → (𝐶 = 𝐴 ↔ 𝐶 = 𝐶))
9		fliftval.5	. . . . . . 7 ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷)
10	9	eqeq2d 2747	. . . . . 6 ⊢ (𝑥 = 𝑌 → (𝐷 = 𝐵 ↔ 𝐷 = 𝐷))
11	8, 10	anbi12d 631	. . . . 5 ⊢ (𝑥 = 𝑌 → ((𝐶 = 𝐴 ∧ 𝐷 = 𝐵) ↔ (𝐶 = 𝐶 ∧ 𝐷 = 𝐷)))
12	11	rspcev 3570	. . . 4 ⊢ ((𝑌 ∈ 𝑋 ∧ (𝐶 = 𝐶 ∧ 𝐷 = 𝐷)) → ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))
13	3, 6, 12	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))
14		flift.1	. . . . 5 ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩)
15		flift.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅)
16		flift.3	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆)
17	14, 15, 16	fliftel 7236	. . . 4 ⊢ (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵)))
18	17	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → (𝐶𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵)))
19	13, 18	mpbird 256	. 2 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → 𝐶𝐹𝐷)
20		funbrfv 6876	. 2 ⊢ (Fun 𝐹 → (𝐶𝐹𝐷 → (𝐹‘𝐶) = 𝐷))
21	2, 19, 20	sylc 65	1 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → (𝐹‘𝐶) = 𝐷)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1540   ∈ wcel 2105  ∃wrex 3070  ⟨cop 4579   class class class wbr 5092   ↦ cmpt 5175  ran crn 5621  Fun wfun 6473  ‘cfv 6479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-iota 6431  df-fun 6481  df-fv 6487

This theorem is referenced by:  qliftval  8666  cygznlem2  20882  pi1xfrval  24323  pi1coval  24329