Theorem fvreseq1 7041

Description: Equality of a function restricted to the domain of another function. (Contributed by AV, 6-Jan-2019.)

Assertion

Ref	Expression
fvreseq1	⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ 𝐵 ⊆ 𝐴) → ((𝐹 ↾ 𝐵) = 𝐺 ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))

Distinct variable groups:   𝑥,𝐵   𝑥,𝐹   𝑥,𝐺

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem fvreseq1

Step	Hyp	Ref	Expression
1		fnresdm 6661	. . . . 5 ⊢ (𝐺 Fn 𝐵 → (𝐺 ↾ 𝐵) = 𝐺)
2	1	ad2antlr 740	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ 𝐵 ⊆ 𝐴) → (𝐺 ↾ 𝐵) = 𝐺)
3	2	eqcomd 2772	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ 𝐵 ⊆ 𝐴) → 𝐺 = (𝐺 ↾ 𝐵))
4	3	eqeq2d 2777	. 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ 𝐵 ⊆ 𝐴) → ((𝐹 ↾ 𝐵) = 𝐺 ↔ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)))
5		ssid 3962	. . 3 ⊢ 𝐵 ⊆ 𝐵
6		fvreseq0 7040	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐵)) → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))
7	5, 6	mpanr2 717	. 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ 𝐵 ⊆ 𝐴) → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))
8	4, 7	bitrd 282	1 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ 𝐵 ⊆ 𝐴) → ((𝐹 ↾ 𝐵) = 𝐺 ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∀wral 3082   ⊆ wss 3908   ↾ cres 5668   Fn wfn 6538  ‘cfv 6543

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-fv 6551

This theorem is used by:  symgextres  19526  ressply1evl  22567  sseqfres  34815  imaidfu  49929  imasubc  49970