Theorem eqfunresadj 7338

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 5979	. 2 ⊢ Rel (𝐹 ↾ (𝑋 ∪ {𝑌}))
2		relres 5979	. 2 ⊢ Rel (𝐺 ↾ (𝑋 ∪ {𝑌}))
3		breq 5112	. . . . 5 ⊢ ((𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) → (𝑥(𝐹 ↾ 𝑋)𝑦 ↔ 𝑥(𝐺 ↾ 𝑋)𝑦))
4	3	3ad2ant2 1134	. . . 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 4608	. . . . . . 7 ⊢ (𝑥 ∈ {𝑌} ↔ 𝑥 = 𝑌)
6		simp33 1212	. . . . . . . . . 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 1198	. . . . . . . . . 10 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑌 ∈ dom 𝐹 ∧ 𝑌 ∈ dom 𝐺 ∧ (𝐹‘𝑌) = (𝐺‘𝑌))) → Fun 𝐹)
9		simp31 1210	. . . . . . . . . 10 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑌 ∈ dom 𝐹 ∧ 𝑌 ∈ dom 𝐺 ∧ (𝐹‘𝑌) = (𝐺‘𝑌))) → 𝑌 ∈ dom 𝐹)
10		funbrfvb 6917	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑌 ∈ dom 𝐹) → ((𝐹‘𝑌) = 𝑦 ↔ 𝑌𝐹𝑦))
11	8, 9, 10	syl2anc 584	. . . . . . . . 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 1199	. . . . . . . . . 10 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑌 ∈ dom 𝐹 ∧ 𝑌 ∈ dom 𝐺 ∧ (𝐹‘𝑌) = (𝐺‘𝑌))) → Fun 𝐺)
13		simp32 1211	. . . . . . . . . 10 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑌 ∈ dom 𝐹 ∧ 𝑌 ∈ dom 𝐺 ∧ (𝐹‘𝑌) = (𝐺‘𝑌))) → 𝑌 ∈ dom 𝐺)
14		funbrfvb 6917	. . . . . . . . . 10 ⊢ ((Fun 𝐺 ∧ 𝑌 ∈ dom 𝐺) → ((𝐺‘𝑌) = 𝑦 ↔ 𝑌𝐺𝑦))
15	12, 13, 14	syl2anc 584	. . . . . . . . 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 309	. . . . . . . 8 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑌 ∈ dom 𝐹 ∧ 𝑌 ∈ dom 𝐺 ∧ (𝐹‘𝑌) = (𝐺‘𝑌))) → (𝑌𝐹𝑦 ↔ 𝑌𝐺𝑦))
17		breq1 5113	. . . . . . . . 9 ⊢ (𝑥 = 𝑌 → (𝑥𝐹𝑦 ↔ 𝑌𝐹𝑦))
18		breq1 5113	. . . . . . . . 9 ⊢ (𝑥 = 𝑌 → (𝑥𝐺𝑦 ↔ 𝑌𝐺𝑦))
19	17, 18	bibi12d 345	. . . . . . . 8 ⊢ (𝑥 = 𝑌 → ((𝑥𝐹𝑦 ↔ 𝑥𝐺𝑦) ↔ (𝑌𝐹𝑦 ↔ 𝑌𝐺𝑦)))
20	16, 19	syl5ibrcom 247	. . . . . . 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 242	. . . . . 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 577	. . . . 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 3454	. . . . . 6 ⊢ 𝑦 ∈ V
24	23	brresi 5962	. . . . 5 ⊢ (𝑥(𝐹 ↾ {𝑌})𝑦 ↔ (𝑥 ∈ {𝑌} ∧ 𝑥𝐹𝑦))
25	23	brresi 5962	. . . . 5 ⊢ (𝑥(𝐺 ↾ {𝑌})𝑦 ↔ (𝑥 ∈ {𝑌} ∧ 𝑥𝐺𝑦))
26	22, 24, 25	3bitr4g 314	. . . 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 918	. . 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 5967	. . . . 5 ⊢ (𝐹 ↾ (𝑋 ∪ {𝑌})) = ((𝐹 ↾ 𝑋) ∪ (𝐹 ↾ {𝑌}))
29	28	breqi 5116	. . . 4 ⊢ (𝑥(𝐹 ↾ (𝑋 ∪ {𝑌}))𝑦 ↔ 𝑥((𝐹 ↾ 𝑋) ∪ (𝐹 ↾ {𝑌}))𝑦)
30		brun 5161	. . . 4 ⊢ (𝑥((𝐹 ↾ 𝑋) ∪ (𝐹 ↾ {𝑌}))𝑦 ↔ (𝑥(𝐹 ↾ 𝑋)𝑦 ∨ 𝑥(𝐹 ↾ {𝑌})𝑦))
31	29, 30	bitri 275	. . 3 ⊢ (𝑥(𝐹 ↾ (𝑋 ∪ {𝑌}))𝑦 ↔ (𝑥(𝐹 ↾ 𝑋)𝑦 ∨ 𝑥(𝐹 ↾ {𝑌})𝑦))
32		resundi 5967	. . . . 5 ⊢ (𝐺 ↾ (𝑋 ∪ {𝑌})) = ((𝐺 ↾ 𝑋) ∪ (𝐺 ↾ {𝑌}))
33	32	breqi 5116	. . . 4 ⊢ (𝑥(𝐺 ↾ (𝑋 ∪ {𝑌}))𝑦 ↔ 𝑥((𝐺 ↾ 𝑋) ∪ (𝐺 ↾ {𝑌}))𝑦)
34		brun 5161	. . . 4 ⊢ (𝑥((𝐺 ↾ 𝑋) ∪ (𝐺 ↾ {𝑌}))𝑦 ↔ (𝑥(𝐺 ↾ 𝑋)𝑦 ∨ 𝑥(𝐺 ↾ {𝑌})𝑦))
35	33, 34	bitri 275	. . 3 ⊢ (𝑥(𝐺 ↾ (𝑋 ∪ {𝑌}))𝑦 ↔ (𝑥(𝐺 ↾ 𝑋)𝑦 ∨ 𝑥(𝐺 ↾ {𝑌})𝑦))
36	27, 31, 35	3bitr4g 314	. 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 5760	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 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ∪ cun 3915  {csn 4592   class class class wbr 5110  dom cdm 5641   ↾ cres 5643  Fun wfun 6508  ‘cfv 6514

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-res 5653  df-iota 6467  df-fun 6516  df-fn 6517  df-fv 6522

This theorem is referenced by:  eqfunressuc  7339