Theorem fvimacnvALT 7034

Description: Alternate proof of fvimacnv 7030, based on funimass3 7031. If funimass3 7031 is ever proved directly, as opposed to using funimacnv 6598 pointwise, then the proof of funimacnv 6598 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 4743	. . 3 ⊢ (𝐴 ∈ dom 𝐹 → {𝐴} ⊆ dom 𝐹)
2		funimass3 7031	. . 3 ⊢ ((Fun 𝐹 ∧ {𝐴} ⊆ dom 𝐹) → ((𝐹 “ {𝐴}) ⊆ 𝐵 ↔ {𝐴} ⊆ (◡𝐹 “ 𝐵)))
3	1, 2	sylan2 602	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹 “ {𝐴}) ⊆ 𝐵 ↔ {𝐴} ⊆ (◡𝐹 “ 𝐵)))
4		fvex 6876	. . . 4 ⊢ (𝐹‘𝐴) ∈ V
5	4	snss 4742	. . 3 ⊢ ((𝐹‘𝐴) ∈ 𝐵 ↔ {(𝐹‘𝐴)} ⊆ 𝐵)
6		eqid 2761	. . . . . 6 ⊢ dom 𝐹 = dom 𝐹
7		df-fn 6520	. . . . . . 7 ⊢ (𝐹 Fn dom 𝐹 ↔ (Fun 𝐹 ∧ dom 𝐹 = dom 𝐹))
8	7	biimpri 230	. . . . . 6 ⊢ ((Fun 𝐹 ∧ dom 𝐹 = dom 𝐹) → 𝐹 Fn dom 𝐹)
9	6, 8	mpan2 701	. . . . 5 ⊢ (Fun 𝐹 → 𝐹 Fn dom 𝐹)
10		fnsnfv 6942	. . . . 5 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐴 ∈ dom 𝐹) → {(𝐹‘𝐴)} = (𝐹 “ {𝐴}))
11	9, 10	sylan 589	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → {(𝐹‘𝐴)} = (𝐹 “ {𝐴}))
12	11	sseq1d 3967	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ({(𝐹‘𝐴)} ⊆ 𝐵 ↔ (𝐹 “ {𝐴}) ⊆ 𝐵))
13	5, 12	bitrid 285	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 ↔ (𝐹 “ {𝐴}) ⊆ 𝐵))
14		snssg 4741	. . 3 ⊢ (𝐴 ∈ dom 𝐹 → (𝐴 ∈ (◡𝐹 “ 𝐵) ↔ {𝐴} ⊆ (◡𝐹 “ 𝐵)))
15	14	adantl 485	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐴 ∈ (◡𝐹 “ 𝐵) ↔ {𝐴} ⊆ (◡𝐹 “ 𝐵)))
16	3, 13, 15	3bitr4d 313	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141   ⊆ wss 3904  {csn 4581  ^◡ccnv 5644  dom cdm 5645   “ cima 5648  Fun wfun 6511   Fn wfn 6512  ‘cfv 6517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-fv 6525

This theorem is referenced by: (None)