Theorem fvimacnvALT 7047

Description: Alternate proof of fvimacnv 7043, based on funimass3 7044. If funimass3 7044 is ever proved directly, as opposed to using funimacnv 6617 pointwise, then the proof of funimacnv 6617 should be replaced with this one. (Contributed by Raph Levien, 20-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
fvimacnvALT	⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵)))

Proof of Theorem fvimacnvALT

Step	Hyp	Ref	Expression
1		snssi 4784	. . 3 ⊢ (𝐴 ∈ dom 𝐹 → {𝐴} ⊆ dom 𝐹)
2		funimass3 7044	. . 3 ⊢ ((Fun 𝐹 ∧ {𝐴} ⊆ dom 𝐹) → ((𝐹 “ {𝐴}) ⊆ 𝐵 ↔ {𝐴} ⊆ (◡𝐹 “ 𝐵)))
3	1, 2	sylan2 593	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹 “ {𝐴}) ⊆ 𝐵 ↔ {𝐴} ⊆ (◡𝐹 “ 𝐵)))
4		fvex 6889	. . . 4 ⊢ (𝐹‘𝐴) ∈ V
5	4	snss 4761	. . 3 ⊢ ((𝐹‘𝐴) ∈ 𝐵 ↔ {(𝐹‘𝐴)} ⊆ 𝐵)
6		eqid 2735	. . . . . 6 ⊢ dom 𝐹 = dom 𝐹
7		df-fn 6534	. . . . . . 7 ⊢ (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
8	7	biimpri 228	. . . . . 6 ⊢ ((Fun 𝐹 ∧ dom 𝐹 = dom 𝐹) → 𝐹 Fn dom 𝐹)
9	6, 8	mpan2 691	. . . . 5 ⊢ (Fun 𝐹 → 𝐹 Fn dom 𝐹)
10		fnsnfv 6958	. . . . 5 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐴 ∈ dom 𝐹) → {(𝐹‘𝐴)} = (𝐹 “ {𝐴}))
11	9, 10	sylan 580	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → {(𝐹‘𝐴)} = (𝐹 “ {𝐴}))
12	11	sseq1d 3990	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ({(𝐹‘𝐴)} ⊆ 𝐵 ↔ (𝐹 “ {𝐴}) ⊆ 𝐵))
13	5, 12	bitrid 283	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 ↔ (𝐹 “ {𝐴}) ⊆ 𝐵))
14		snssg 4759	. . 3 ⊢ (𝐴 ∈ dom 𝐹 → (𝐴 ∈ (◡𝐹 “ 𝐵) ↔ {𝐴} ⊆ (◡𝐹 “ 𝐵)))
15	14	adantl 481	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐴 ∈ (◡𝐹 “ 𝐵) ↔ {𝐴} ⊆ (◡𝐹 “ 𝐵)))
16	3, 13, 15	3bitr4d 311	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ⊆ wss 3926  {csn 4601  ^◡ccnv 5653  dom cdm 5654   “ cima 5657  Fun wfun 6525   Fn wfn 6526  ‘cfv 6531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-fv 6539

This theorem is referenced by: (None)