Theorem eqfunresadj 7359

Description: Law for adjoining an element to restrictions of functions. (Contributed by Scott Fenton, 6-Dec-2021.)

Assertion

Ref	Expression
eqfunresadj	⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑌 ∈ dom 𝐹 ∧ 𝑌 ∈ dom 𝐺 ∧ (𝐹‘𝑌) = (𝐺‘𝑌))) → (𝐹 ↾ (𝑋 ∪ {𝑌})) = (𝐺 ↾ (𝑋 ∪ {𝑌})))

Proof of Theorem eqfunresadj

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relres 6009	. 2 ⊢ Rel (𝐹 ↾ (𝑋 ∪ {𝑌}))
2		relres 6009	. 2 ⊢ Rel (𝐺 ↾ (𝑋 ∪ {𝑌}))
3		breq 5149	. . . . 5 ⊢ ((𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) → (𝑥(𝐹 ↾ 𝑋)𝑦 ↔ 𝑥(𝐺 ↾ 𝑋)𝑦))
4	3	3ad2ant2 1132	. . . 4 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑌 ∈ dom 𝐹 ∧ 𝑌 ∈ dom 𝐺 ∧ (𝐹‘𝑌) = (𝐺‘𝑌))) → (𝑥(𝐹 ↾ 𝑋)𝑦 ↔ 𝑥(𝐺 ↾ 𝑋)𝑦))
5		velsn 4643	. . . . . . 7 ⊢ (𝑥 ∈ {𝑌} ↔ 𝑥 = 𝑌)
6		simp33 1209	. . . . . . . . . 10 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑌 ∈ dom 𝐹 ∧ 𝑌 ∈ dom 𝐺 ∧ (𝐹‘𝑌) = (𝐺‘𝑌))) → (𝐹‘𝑌) = (𝐺‘𝑌))
7	6	eqeq1d 2732	. . . . . . . . 9 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑌 ∈ dom 𝐹 ∧ 𝑌 ∈ dom 𝐺 ∧ (𝐹‘𝑌) = (𝐺‘𝑌))) → ((𝐹‘𝑌) = 𝑦 ↔ (𝐺‘𝑌) = 𝑦))
8		simp1l 1195	. . . . . . . . . 10 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑌 ∈ dom 𝐹 ∧ 𝑌 ∈ dom 𝐺 ∧ (𝐹‘𝑌) = (𝐺‘𝑌))) → Fun 𝐹)
9		simp31 1207	. . . . . . . . . 10 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑌 ∈ dom 𝐹 ∧ 𝑌 ∈ dom 𝐺 ∧ (𝐹‘𝑌) = (𝐺‘𝑌))) → 𝑌 ∈ dom 𝐹)
10		funbrfvb 6945	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑌 ∈ dom 𝐹) → ((𝐹‘𝑌) = 𝑦 ↔ 𝑌𝐹𝑦))
11	8, 9, 10	syl2anc 582	. . . . . . . . 9 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑌 ∈ dom 𝐹 ∧ 𝑌 ∈ dom 𝐺 ∧ (𝐹‘𝑌) = (𝐺‘𝑌))) → ((𝐹‘𝑌) = 𝑦 ↔ 𝑌𝐹𝑦))
12		simp1r 1196	. . . . . . . . . 10 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑌 ∈ dom 𝐹 ∧ 𝑌 ∈ dom 𝐺 ∧ (𝐹‘𝑌) = (𝐺‘𝑌))) → Fun 𝐺)
13		simp32 1208	. . . . . . . . . 10 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑌 ∈ dom 𝐹 ∧ 𝑌 ∈ dom 𝐺 ∧ (𝐹‘𝑌) = (𝐺‘𝑌))) → 𝑌 ∈ dom 𝐺)
14		funbrfvb 6945	. . . . . . . . . 10 ⊢ ((Fun 𝐺 ∧ 𝑌 ∈ dom 𝐺) → ((𝐺‘𝑌) = 𝑦 ↔ 𝑌𝐺𝑦))
15	12, 13, 14	syl2anc 582	. . . . . . . . 9 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑌 ∈ dom 𝐹 ∧ 𝑌 ∈ dom 𝐺 ∧ (𝐹‘𝑌) = (𝐺‘𝑌))) → ((𝐺‘𝑌) = 𝑦 ↔ 𝑌𝐺𝑦))
16	7, 11, 15	3bitr3d 308	. . . . . . . 8 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑌 ∈ dom 𝐹 ∧ 𝑌 ∈ dom 𝐺 ∧ (𝐹‘𝑌) = (𝐺‘𝑌))) → (𝑌𝐹𝑦 ↔ 𝑌𝐺𝑦))
17		breq1 5150	. . . . . . . . 9 ⊢ (𝑥 = 𝑌 → (𝑥𝐹𝑦 ↔ 𝑌𝐹𝑦))
18		breq1 5150	. . . . . . . . 9 ⊢ (𝑥 = 𝑌 → (𝑥𝐺𝑦 ↔ 𝑌𝐺𝑦))
19	17, 18	bibi12d 344	. . . . . . . 8 ⊢ (𝑥 = 𝑌 → ((𝑥𝐹𝑦 ↔ 𝑥𝐺𝑦) ↔ (𝑌𝐹𝑦 ↔ 𝑌𝐺𝑦)))
20	16, 19	syl5ibrcom 246	. . . . . . 7 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑌 ∈ dom 𝐹 ∧ 𝑌 ∈ dom 𝐺 ∧ (𝐹‘𝑌) = (𝐺‘𝑌))) → (𝑥 = 𝑌 → (𝑥𝐹𝑦 ↔ 𝑥𝐺𝑦)))
21	5, 20	biimtrid 241	. . . . . 6 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑌 ∈ dom 𝐹 ∧ 𝑌 ∈ dom 𝐺 ∧ (𝐹‘𝑌) = (𝐺‘𝑌))) → (𝑥 ∈ {𝑌} → (𝑥𝐹𝑦 ↔ 𝑥𝐺𝑦)))
22	21	pm5.32d 575	. . . . 5 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑌 ∈ dom 𝐹 ∧ 𝑌 ∈ dom 𝐺 ∧ (𝐹‘𝑌) = (𝐺‘𝑌))) → ((𝑥 ∈ {𝑌} ∧ 𝑥𝐹𝑦) ↔ (𝑥 ∈ {𝑌} ∧ 𝑥𝐺𝑦)))
23		vex 3476	. . . . . 6 ⊢ 𝑦 ∈ V
24	23	brresi 5989	. . . . 5 ⊢ (𝑥(𝐹 ↾ {𝑌})𝑦 ↔ (𝑥 ∈ {𝑌} ∧ 𝑥𝐹𝑦))
25	23	brresi 5989	. . . . 5 ⊢ (𝑥(𝐺 ↾ {𝑌})𝑦 ↔ (𝑥 ∈ {𝑌} ∧ 𝑥𝐺𝑦))
26	22, 24, 25	3bitr4g 313	. . . 4 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑌 ∈ dom 𝐹 ∧ 𝑌 ∈ dom 𝐺 ∧ (𝐹‘𝑌) = (𝐺‘𝑌))) → (𝑥(𝐹 ↾ {𝑌})𝑦 ↔ 𝑥(𝐺 ↾ {𝑌})𝑦))
27	4, 26	orbi12d 915	. . 3 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑌 ∈ dom 𝐹 ∧ 𝑌 ∈ dom 𝐺 ∧ (𝐹‘𝑌) = (𝐺‘𝑌))) → ((𝑥(𝐹 ↾ 𝑋)𝑦 ∨ 𝑥(𝐹 ↾ {𝑌})𝑦) ↔ (𝑥(𝐺 ↾ 𝑋)𝑦 ∨ 𝑥(𝐺 ↾ {𝑌})𝑦)))
28		resundi 5994	. . . . 5 ⊢ (𝐹 ↾ (𝑋 ∪ {𝑌})) = ((𝐹 ↾ 𝑋) ∪ (𝐹 ↾ {𝑌}))
29	28	breqi 5153	. . . 4 ⊢ (𝑥(𝐹 ↾ (𝑋 ∪ {𝑌}))𝑦 ↔ 𝑥((𝐹 ↾ 𝑋) ∪ (𝐹 ↾ {𝑌}))𝑦)
30		brun 5198	. . . 4 ⊢ (𝑥((𝐹 ↾ 𝑋) ∪ (𝐹 ↾ {𝑌}))𝑦 ↔ (𝑥(𝐹 ↾ 𝑋)𝑦 ∨ 𝑥(𝐹 ↾ {𝑌})𝑦))
31	29, 30	bitri 274	. . 3 ⊢ (𝑥(𝐹 ↾ (𝑋 ∪ {𝑌}))𝑦 ↔ (𝑥(𝐹 ↾ 𝑋)𝑦 ∨ 𝑥(𝐹 ↾ {𝑌})𝑦))
32		resundi 5994	. . . . 5 ⊢ (𝐺 ↾ (𝑋 ∪ {𝑌})) = ((𝐺 ↾ 𝑋) ∪ (𝐺 ↾ {𝑌}))
33	32	breqi 5153	. . . 4 ⊢ (𝑥(𝐺 ↾ (𝑋 ∪ {𝑌}))𝑦 ↔ 𝑥((𝐺 ↾ 𝑋) ∪ (𝐺 ↾ {𝑌}))𝑦)
34		brun 5198	. . . 4 ⊢ (𝑥((𝐺 ↾ 𝑋) ∪ (𝐺 ↾ {𝑌}))𝑦 ↔ (𝑥(𝐺 ↾ 𝑋)𝑦 ∨ 𝑥(𝐺 ↾ {𝑌})𝑦))
35	33, 34	bitri 274	. . 3 ⊢ (𝑥(𝐺 ↾ (𝑋 ∪ {𝑌}))𝑦 ↔ (𝑥(𝐺 ↾ 𝑋)𝑦 ∨ 𝑥(𝐺 ↾ {𝑌})𝑦))
36	27, 31, 35	3bitr4g 313	. 2 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑌 ∈ dom 𝐹 ∧ 𝑌 ∈ dom 𝐺 ∧ (𝐹‘𝑌) = (𝐺‘𝑌))) → (𝑥(𝐹 ↾ (𝑋 ∪ {𝑌}))𝑦 ↔ 𝑥(𝐺 ↾ (𝑋 ∪ {𝑌}))𝑦))
37	1, 2, 36	eqbrrdiv 5793	1 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑌 ∈ dom 𝐹 ∧ 𝑌 ∈ dom 𝐺 ∧ (𝐹‘𝑌) = (𝐺‘𝑌))) → (𝐹 ↾ (𝑋 ∪ {𝑌})) = (𝐺 ↾ (𝑋 ∪ {𝑌})))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   ∪ cun 3945  {csn 4627   class class class wbr 5147  dom cdm 5675   ↾ cres 5677  Fun wfun 6536  ‘cfv 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-res 5687  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550

This theorem is referenced by:  eqfunressuc  7360