Theorem fvimacnvALT 7003

Description: Alternate proof of fvimacnv 6999, based on funimass3 7000. If funimass3 7000 is ever proved directly, as opposed to using funimacnv 6573 pointwise, then the proof of funimacnv 6573 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 4752	. . 3 ⊢ (𝐴 ∈ dom 𝐹 → {𝐴} ⊆ dom 𝐹)
2		funimass3 7000	. . 3 ⊢ ((Fun 𝐹 ∧ {𝐴} ⊆ dom 𝐹) → ((𝐹 “ {𝐴}) ⊆ 𝐵 ↔ {𝐴} ⊆ (◡𝐹 “ 𝐵)))
3	1, 2	sylan2 594	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹 “ {𝐴}) ⊆ 𝐵 ↔ {𝐴} ⊆ (◡𝐹 “ 𝐵)))
4		fvex 6847	. . . 4 ⊢ (𝐹‘𝐴) ∈ V
5	4	snss 4729	. . 3 ⊢ ((𝐹‘𝐴) ∈ 𝐵 ↔ {(𝐹‘𝐴)} ⊆ 𝐵)
6		eqid 2737	. . . . . 6 ⊢ dom 𝐹 = dom 𝐹
7		df-fn 6495	. . . . . . 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 692	. . . . 5 ⊢ (Fun 𝐹 → 𝐹 Fn dom 𝐹)
10		fnsnfv 6913	. . . . 5 ⊢ ((𝐹 Fn dom 𝐹 ∧ 𝐴 ∈ dom 𝐹) → {(𝐹‘𝐴)} = (𝐹 “ {𝐴}))
11	9, 10	sylan 581	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → {(𝐹‘𝐴)} = (𝐹 “ {𝐴}))
12	11	sseq1d 3954	. . 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 4728	. . 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 1542   ∈ wcel 2114   ⊆ wss 3890  {csn 4568  ^◡ccnv 5623  dom cdm 5624   “ cima 5627  Fun wfun 6486   Fn wfn 6487  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500

This theorem is referenced by: (None)