Theorem imainss 5183

Description: An upper bound for intersection with an image. Theorem 41 of [Suppes] p. 66. (Contributed by NM, 11-Aug-2004.)

Assertion

Ref	Expression
imainss	⊢ ((𝑅 “ 𝐴) ∩ 𝐵) ⊆ (𝑅 “ (𝐴 ∩ (◡𝑅 “ 𝐵)))

Proof of Theorem imainss

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

Step	Hyp	Ref	Expression
1		vex 2818	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
2		vex 2818	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
3	1, 2	brcnv 4943	. . . . . . . . . 10 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
4		19.8a 1639	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥) → ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥))
5	3, 4	sylan2br 288	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥𝑅𝑦) → ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥))
6	5	ancoms 268	. . . . . . . 8 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦 ∈ 𝐵) → ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥))
7	6	anim2i 342	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥𝑅𝑦 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥)))
8		simprl 531	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥𝑅𝑦 ∧ 𝑦 ∈ 𝐵)) → 𝑥𝑅𝑦)
9	7, 8	jca 306	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥𝑅𝑦 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥)) ∧ 𝑥𝑅𝑦))
10	9	anassrs 400	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥)) ∧ 𝑥𝑅𝑦))
11		elin 3406	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ (◡𝑅 “ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (◡𝑅 “ 𝐵)))
12	2	elima2 5112	. . . . . . . 8 ⊢ (𝑥 ∈ (◡𝑅 “ 𝐵) ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥))
13	12	anbi2i 457	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ (◡𝑅 “ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥)))
14	11, 13	bitri 184	. . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ (◡𝑅 “ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥)))
15	14	anbi1i 458	. . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∩ (◡𝑅 “ 𝐵)) ∧ 𝑥𝑅𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦◡𝑅𝑥)) ∧ 𝑥𝑅𝑦))
16	10, 15	sylibr 134	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ (𝐴 ∩ (◡𝑅 “ 𝐵)) ∧ 𝑥𝑅𝑦))
17	16	eximi 1649	. . 3 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵) → ∃𝑥(𝑥 ∈ (𝐴 ∩ (◡𝑅 “ 𝐵)) ∧ 𝑥𝑅𝑦))
18	1	elima2 5112	. . . . 5 ⊢ (𝑦 ∈ (𝑅 “ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦))
19	18	anbi1i 458	. . . 4 ⊢ ((𝑦 ∈ (𝑅 “ 𝐴) ∧ 𝑦 ∈ 𝐵) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵))
20		elin 3406	. . . 4 ⊢ (𝑦 ∈ ((𝑅 “ 𝐴) ∩ 𝐵) ↔ (𝑦 ∈ (𝑅 “ 𝐴) ∧ 𝑦 ∈ 𝐵))
21		19.41v 1954	. . . 4 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵))
22	19, 20, 21	3bitr4i 212	. . 3 ⊢ (𝑦 ∈ ((𝑅 “ 𝐴) ∩ 𝐵) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝑦) ∧ 𝑦 ∈ 𝐵))
23	1	elima2 5112	. . 3 ⊢ (𝑦 ∈ (𝑅 “ (𝐴 ∩ (◡𝑅 “ 𝐵))) ↔ ∃𝑥(𝑥 ∈ (𝐴 ∩ (◡𝑅 “ 𝐵)) ∧ 𝑥𝑅𝑦))
24	17, 22, 23	3imtr4i 201	. 2 ⊢ (𝑦 ∈ ((𝑅 “ 𝐴) ∩ 𝐵) → 𝑦 ∈ (𝑅 “ (𝐴 ∩ (◡𝑅 “ 𝐵))))
25	24	ssriv 3246	1 ⊢ ((𝑅 “ 𝐴) ∩ 𝐵) ⊆ (𝑅 “ (𝐴 ∩ (◡𝑅 “ 𝐵)))

Syntax hints:   ∧ wa 104  ∃wex 1541   ∈ wcel 2205   ∩ cin 3213   ⊆ wss 3214   class class class wbr 4114  ^◡ccnv 4753   “ cima 4757

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-xp 4760  df-cnv 4762  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767

This theorem is referenced by: (None)