Theorem eqfunresadj 7294

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 5953	. 2 ⊢ Rel (𝐹 ↾ (𝑋 ∪ {𝑌}))
2		relres 5953	. 2 ⊢ Rel (𝐺 ↾ (𝑋 ∪ {𝑌}))
3		breq 5091	. . . . 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 4589	. . . . . . 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 2733	. . . . . . . . 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 6875	. . . . . . . . . 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 6875	. . . . . . . . . 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 5092	. . . . . . . . 9 ⊢ (𝑥 = 𝑌 → (𝑥𝐹𝑦 ↔ 𝑌𝐹𝑦))
18		breq1 5092	. . . . . . . . 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 3440	. . . . . 6 ⊢ 𝑦 ∈ V
24	23	brresi 5936	. . . . 5 ⊢ (𝑥(𝐹 ↾ {𝑌})𝑦 ↔ (𝑥 ∈ {𝑌} ∧ 𝑥𝐹𝑦))
25	23	brresi 5936	. . . . 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 5941	. . . . 5 ⊢ (𝐹 ↾ (𝑋 ∪ {𝑌})) = ((𝐹 ↾ 𝑋) ∪ (𝐹 ↾ {𝑌}))
29	28	breqi 5095	. . . 4 ⊢ (𝑥(𝐹 ↾ (𝑋 ∪ {𝑌}))𝑦 ↔ 𝑥((𝐹 ↾ 𝑋) ∪ (𝐹 ↾ {𝑌}))𝑦)
30		brun 5140	. . . 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 5941	. . . . 5 ⊢ (𝐺 ↾ (𝑋 ∪ {𝑌})) = ((𝐺 ↾ 𝑋) ∪ (𝐺 ↾ {𝑌}))
33	32	breqi 5095	. . . 4 ⊢ (𝑥(𝐺 ↾ (𝑋 ∪ {𝑌}))𝑦 ↔ 𝑥((𝐺 ↾ 𝑋) ∪ (𝐺 ↾ {𝑌}))𝑦)
34		brun 5140	. . . 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 5733	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 1541   ∈ wcel 2111   ∪ cun 3895  {csn 4573   class class class wbr 5089  dom cdm 5614   ↾ cres 5616  Fun wfun 6475  ‘cfv 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-res 5626  df-iota 6437  df-fun 6483  df-fn 6484  df-fv 6489

This theorem is referenced by:  eqfunressuc  7295