Theorem ralima 5837

Description: Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Hypothesis

Ref	Expression
rexima.x	⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ralima	⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓))

Distinct variable groups:   𝜑,𝑦   𝜓,𝑥   𝑥,𝐹,𝑦   𝑥,𝐵,𝑦   𝑥,𝐴,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem ralima

Step	Hyp	Ref	Expression
1		ssel2 3192	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐴)
2		funfvex 5606	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝐹‘𝑦) ∈ V)
3	2	funfni 5385	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ V)
4	1, 3	sylan2 286	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘𝑦) ∈ V)
5	4	anassrs 400	. 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝐹‘𝑦) ∈ V)
6		fvelimab 5648	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ (𝐹 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥))
7		eqcom 2208	. . . 4 ⊢ ((𝐹‘𝑦) = 𝑥 ↔ 𝑥 = (𝐹‘𝑦))
8	7	rexbii 2514	. . 3 ⊢ (∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑥 ↔ ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦))
9	6, 8	bitrdi 196	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑥 ∈ (𝐹 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 𝑥 = (𝐹‘𝑦)))
10		rexima.x	. . 3 ⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))
11	10	adantl 277	. 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑥 = (𝐹‘𝑦)) → (𝜑 ↔ 𝜓))
12	5, 9, 11	ralxfr2d 4519	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373   ∈ wcel 2177  ∀wral 2485  ∃wrex 2486  Vcvv 2773   ⊆ wss 3170   “ cima 4686   Fn wfn 5275  ‘cfv 5280

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-fv 5288

This theorem is referenced by:  supisolem  7125  mhmima  13398  ghmnsgima  13679  qtopbasss  15068  fsumdvdsmul  15538