Theorem imasetpreimafvbijlemfv 46528

Description: Lemma for imasetpreimafvbij 46532: the value of the mapping 𝐻 at a preimage of a value of function 𝐹. (Contributed by AV, 5-Mar-2024.)

Hypotheses

Ref	Expression
fundcmpsurinj.p	⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
fundcmpsurinj.h	⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))

Assertion

Ref	Expression
imasetpreimafvbijlemfv	⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → (𝐻‘𝑌) = (𝐹‘𝑋))

Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧,𝑝   𝑃,𝑝   𝑋,𝑝   𝐴,𝑝,𝑥,𝑧   𝑥,𝑃   𝑥,𝑋   𝑌,𝑝,𝑥,𝑧

Allowed substitution hints:   𝑃(𝑧)   𝐻(𝑥,𝑧,𝑝)   𝑋(𝑧)

Proof of Theorem imasetpreimafvbijlemfv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnfun 6649	. . . . 5 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2	1	anim1i 614	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃) → (Fun 𝐹 ∧ 𝑌 ∈ 𝑃))
3	2	3adant3 1131	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → (Fun 𝐹 ∧ 𝑌 ∈ 𝑃))
4		fundcmpsurinj.p	. . . 4 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
5		fundcmpsurinj.h	. . . 4 ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))
6	4, 5	fundcmpsurinjlem3 46526	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑌 ∈ 𝑃) → (𝐻‘𝑌) = ∪ (𝐹 “ 𝑌))
7	3, 6	syl 17	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → (𝐻‘𝑌) = ∪ (𝐹 “ 𝑌))
8	1	3ad2ant1 1132	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → Fun 𝐹)
9		funiunfv 7250	. . 3 ⊢ (Fun 𝐹 → ∪ 𝑦 ∈ 𝑌 (𝐹‘𝑦) = ∪ (𝐹 “ 𝑌))
10	8, 9	syl 17	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → ∪ 𝑦 ∈ 𝑌 (𝐹‘𝑦) = ∪ (𝐹 “ 𝑌))
11		simp3 1137	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → 𝑋 ∈ 𝑌)
12		simpl1 1190	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) ∧ 𝑦 ∈ 𝑌) → 𝐹 Fn 𝐴)
13		simpl2 1191	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) ∧ 𝑦 ∈ 𝑌) → 𝑌 ∈ 𝑃)
14		simpr 484	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌)
15		simpl3 1192	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) ∧ 𝑦 ∈ 𝑌) → 𝑋 ∈ 𝑌)
16	4	elsetpreimafveqfv 46518	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑌 ∈ 𝑃 ∧ 𝑦 ∈ 𝑌 ∧ 𝑋 ∈ 𝑌)) → (𝐹‘𝑦) = (𝐹‘𝑋))
17	12, 13, 14, 15, 16	syl13anc 1371	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) ∧ 𝑦 ∈ 𝑌) → (𝐹‘𝑦) = (𝐹‘𝑋))
18	17	ralrimiva 3145	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → ∀𝑦 ∈ 𝑌 (𝐹‘𝑦) = (𝐹‘𝑋))
19		fveq2 6891	. . . 4 ⊢ (𝑦 = 𝑋 → (𝐹‘𝑦) = (𝐹‘𝑋))
20	19	iuneqconst 5008	. . 3 ⊢ ((𝑋 ∈ 𝑌 ∧ ∀𝑦 ∈ 𝑌 (𝐹‘𝑦) = (𝐹‘𝑋)) → ∪ 𝑦 ∈ 𝑌 (𝐹‘𝑦) = (𝐹‘𝑋))
21	11, 18, 20	syl2anc 583	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → ∪ 𝑦 ∈ 𝑌 (𝐹‘𝑦) = (𝐹‘𝑋))
22	7, 10, 21	3eqtr2d 2777	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → (𝐻‘𝑌) = (𝐹‘𝑋))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  {cab 2708  ∀wral 3060  ∃wrex 3069  {csn 4628  ∪ cuni 4908  ∪ ciun 4997   ↦ cmpt 5231  ^◡ccnv 5675   “ cima 5679  Fun wfun 6537   Fn wfn 6538  ‘cfv 6543

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551

This theorem is referenced by:  imasetpreimafvbijlemfv1  46529