Theorem fniniseg 5776

Description: Membership in the preimage of a singleton, under a function. (Contributed by Mario Carneiro, 12-May-2014.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
fniniseg	⊢ (𝐹 Fn 𝐴 → (𝐶 ∈ (◡𝐹 “ {𝐵}) ↔ (𝐶 ∈ 𝐴 ∧ (𝐹‘𝐶) = 𝐵)))

Proof of Theorem fniniseg

Step	Hyp	Ref	Expression
1		elpreima 5775	. 2 ⊢ (𝐹 Fn 𝐴 → (𝐶 ∈ (◡𝐹 “ {𝐵}) ↔ (𝐶 ∈ 𝐴 ∧ (𝐹‘𝐶) ∈ {𝐵})))
2		funfvex 5665	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐶 ∈ dom 𝐹) → (𝐹‘𝐶) ∈ V)
3		elsng 3688	. . . . 5 ⊢ ((𝐹‘𝐶) ∈ V → ((𝐹‘𝐶) ∈ {𝐵} ↔ (𝐹‘𝐶) = 𝐵))
4	2, 3	syl 14	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐶 ∈ dom 𝐹) → ((𝐹‘𝐶) ∈ {𝐵} ↔ (𝐹‘𝐶) = 𝐵))
5	4	funfni 5439	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ∈ 𝐴) → ((𝐹‘𝐶) ∈ {𝐵} ↔ (𝐹‘𝐶) = 𝐵))
6	5	pm5.32da 452	. 2 ⊢ (𝐹 Fn 𝐴 → ((𝐶 ∈ 𝐴 ∧ (𝐹‘𝐶) ∈ {𝐵}) ↔ (𝐶 ∈ 𝐴 ∧ (𝐹‘𝐶) = 𝐵)))
7	1, 6	bitrd 188	1 ⊢ (𝐹 Fn 𝐴 → (𝐶 ∈ (◡𝐹 “ {𝐵}) ↔ (𝐶 ∈ 𝐴 ∧ (𝐹‘𝐶) = 𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2202  Vcvv 2803  {csn 3673  ^◡ccnv 4730  dom cdm 4731   “ cima 4734  Fun wfun 5327   Fn wfn 5328  ‘cfv 5333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341

This theorem is referenced by:  pw2f1odclem  7063  ghmeqker  13938  pilem1  15590  taupi  16806