Theorem imadiflem 5277

Description: One direction of imadif 5278. This direction does not require Fun ^◡𝐹. (Contributed by Jim Kingdon, 25-Dec-2018.)

Assertion

Ref	Expression
imadiflem	⊢ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) ⊆ (𝐹 “ (𝐴 ∖ 𝐵))

Proof of Theorem imadiflem

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

Step	Hyp	Ref	Expression
1		df-rex 2454	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))
2		df-rex 2454	. . . . 5 ⊢ (∃𝑥 ∈ 𝐵 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
3	2	notbii 663	. . . 4 ⊢ (¬ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦 ↔ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
4		alnex 1492	. . . . . . 7 ⊢ (∀𝑥 ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦) ↔ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
5		19.29r 1614	. . . . . . 7 ⊢ ((∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ∀𝑥 ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
6	4, 5	sylan2br 286	. . . . . 6 ⊢ ((∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
7		simpl 108	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))
8		simplr 525	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → 𝑥𝐹𝑦)
9		simpr 109	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦))
10		ancom 264	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦) ↔ (𝑥𝐹𝑦 ∧ 𝑥 ∈ 𝐵))
11	10	notbii 663	. . . . . . . . . . . 12 ⊢ (¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦) ↔ ¬ (𝑥𝐹𝑦 ∧ 𝑥 ∈ 𝐵))
12		imnan 685	. . . . . . . . . . . 12 ⊢ ((𝑥𝐹𝑦 → ¬ 𝑥 ∈ 𝐵) ↔ ¬ (𝑥𝐹𝑦 ∧ 𝑥 ∈ 𝐵))
13	11, 12	bitr4i 186	. . . . . . . . . . 11 ⊢ (¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦) ↔ (𝑥𝐹𝑦 → ¬ 𝑥 ∈ 𝐵))
14	9, 13	sylib 121	. . . . . . . . . 10 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → (𝑥𝐹𝑦 → ¬ 𝑥 ∈ 𝐵))
15	8, 14	mpd 13	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → ¬ 𝑥 ∈ 𝐵)
16	7, 15, 8	jca32 308	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
17		eldif 3130	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
18	17	anbi1i 455	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦))
19		anandir 586	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
20	18, 19	bitri 183	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥𝐹𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ (¬ 𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)))
21	16, 20	sylibr 133	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥𝐹𝑦))
22	21	eximi 1593	. . . . . 6 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ (𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → ∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥𝐹𝑦))
23	6, 22	syl 14	. . . . 5 ⊢ ((∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → ∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥𝐹𝑦))
24		df-rex 2454	. . . . 5 ⊢ (∃𝑥 ∈ (𝐴 ∖ 𝐵)𝑥𝐹𝑦 ↔ ∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝑥𝐹𝑦))
25	23, 24	sylibr 133	. . . 4 ⊢ ((∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ∧ ¬ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐹𝑦)) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)𝑥𝐹𝑦)
26	1, 3, 25	syl2anb 289	. . 3 ⊢ ((∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)𝑥𝐹𝑦)
27	26	ss2abi 3219	. 2 ⊢ {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦)} ⊆ {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∖ 𝐵)𝑥𝐹𝑦}
28		dfima2 4955	. . . 4 ⊢ (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦}
29		dfima2 4955	. . . 4 ⊢ (𝐹 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦}
30	28, 29	difeq12i 3243	. . 3 ⊢ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) = ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦} ∖ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦})
31		difab 3396	. . 3 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑥𝐹𝑦} ∖ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦}) = {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦)}
32	30, 31	eqtri 2191	. 2 ⊢ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) = {𝑦 ∣ (∃𝑥 ∈ 𝐴 𝑥𝐹𝑦 ∧ ¬ ∃𝑥 ∈ 𝐵 𝑥𝐹𝑦)}
33		dfima2 4955	. 2 ⊢ (𝐹 “ (𝐴 ∖ 𝐵)) = {𝑦 ∣ ∃𝑥 ∈ (𝐴 ∖ 𝐵)𝑥𝐹𝑦}
34	27, 32, 33	3sstr4i 3188	1 ⊢ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) ⊆ (𝐹 “ (𝐴 ∖ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  ∀wal 1346  ∃wex 1485   ∈ wcel 2141  {cab 2156  ∃wrex 2449   ∖ cdif 3118   ⊆ wss 3121   class class class wbr 3989   “ cima 4614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624

This theorem is referenced by:  imadif  5278